Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 67.3% → 96.6%

Time: 8.4s

Alternatives: 17

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
}

Python

def code(x, y, z, t, a, b):
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
}

Python

def code(x, y, z, t, a, b):
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-291}:\\ \;\;\;\;\frac{t\_1}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 10^{+286}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{z - 1} - \frac{a - t}{\frac{y}{z} - \left(y - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- t a) z (* y x)))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma (/ y (fma z (- b y) y)) x (/ (- t a) (- b y)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -4e-291)
       (/ t_1 t_2)
       (if (<= t_3 0.0)
         t_4
         (if (<= t_3 1e+286)
           (/ 1.0 (/ t_2 t_1))
           (- (* x (/ -1.0 (- z 1.0))) (/ (- a t) (- (/ y z) (- y b))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t - a), z, (y * x));
	double t_2 = fma((b - y), z, y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma((y / fma(z, (b - y), y)), x, ((t - a) / (b - y)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -4e-291) {
		tmp = t_1 / t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 1e+286) {
		tmp = 1.0 / (t_2 / t_1);
	} else {
		tmp = (x * (-1.0 / (z - 1.0))) - ((a - t) / ((y / z) - (y - b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(t - a), z, Float64(y * x))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(Float64(y / fma(z, Float64(b - y), y)), x, Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -4e-291)
		tmp = Float64(t_1 / t_2);
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 1e+286)
		tmp = Float64(1.0 / Float64(t_2 / t_1));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / Float64(z - 1.0))) - Float64(Float64(a - t) / Float64(Float64(y / z) - Float64(y - b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -4e-291], N[(t$95$1 / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 1e+286], N[(1.0 / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a - t), $MachinePrecision] / N[(N[(y / z), $MachinePrecision] - N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or -3.99999999999999985e-291 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 76.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   8. Applied rewrites 76.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -3.99999999999999985e-291

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right) + x \cdot y}}{y + z \cdot \left(b - y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right)} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - a\right) \cdot z} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      5. lower-fma.f64 67.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}}{y + z \cdot \left(b - y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}{y + z \cdot \left(b - y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-*.f64 67.3

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{y + z \cdot \left(b - y\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right) + y}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right)} + y} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\left(b - y\right) \cdot z} + y} \]

      13. lower-fma.f64 67.3

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   3. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   if -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000003e286

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      4. lower-/.f64 67.2

         \[\leadsto \frac{1}{\color{blue}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y + z \cdot \left(b - y\right)}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{z \cdot \left(b - y\right) + y}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{z \cdot \left(b - y\right)} + y}{x \cdot y + z \cdot \left(t - a\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - y\right) \cdot z} + y}{x \cdot y + z \cdot \left(t - a\right)}} \]

      9. lower-fma.f64 67.2

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{z \cdot \left(t - a\right) + x \cdot y}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{z \cdot \left(t - a\right)} + x \cdot y}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{\left(t - a\right) \cdot z} + x \cdot y}} \]

      14. lower-fma.f64 67.2

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}} \]

      17. lower-*.f64 67.2

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}} \]

   3. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, y \cdot x\right)}}} \]

   if 1.00000000000000003e286 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

      2. lower--.f64 82.9

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z - \color{blue}{1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{z - 1} \cdot x + \frac{t - a}{\left(b - y\right) + \frac{y}{z}}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\frac{-1}{z - 1} \cdot x - \left(\mathsf{neg}\left(\frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{z - 1} \cdot x - \left(\mathsf{neg}\left(\frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-1}{z - 1}} - \left(\mathsf{neg}\left(\frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{-1}{z - 1}} - \left(\mathsf{neg}\left(\frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)\right) \]

      6. lift--.f64 N/A

         \[\leadsto x \cdot \frac{-1}{z - 1} - \left(\mathsf{neg}\left(\frac{\color{blue}{t - a}}{\left(b - y\right) + \frac{y}{z}}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x \cdot \frac{-1}{z - 1} - \left(\mathsf{neg}\left(\color{blue}{\frac{t - a}{\left(b - y\right) + \frac{y}{z}}}\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto x \cdot \frac{-1}{z - 1} - \color{blue}{\frac{\mathsf{neg}\left(\left(t - a\right)\right)}{\left(b - y\right) + \frac{y}{z}}} \]

      9. lower-/.f64 N/A

         \[\leadsto x \cdot \frac{-1}{z - 1} - \color{blue}{\frac{\mathsf{neg}\left(\left(t - a\right)\right)}{\left(b - y\right) + \frac{y}{z}}} \]

      10. sub-negate-rev N/A

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{\color{blue}{a - t}}{\left(b - y\right) + \frac{y}{z}} \]

      11. lower--.f64 82.9

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{\color{blue}{a - t}}{\left(b - y\right) + \frac{y}{z}} \]

      12. lift-+.f64 N/A

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{a - t}{\color{blue}{\left(b - y\right) + \frac{y}{z}}} \]

      13. +-commutative N/A

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{a - t}{\color{blue}{\frac{y}{z} + \left(b - y\right)}} \]

      14. lift--.f64 N/A

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{a - t}{\frac{y}{z} + \color{blue}{\left(b - y\right)}} \]

      15. sub-negate-rev N/A

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{a - t}{\frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(\left(y - b\right)\right)\right)}} \]

      16. sub-flip-reverse N/A

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{a - t}{\color{blue}{\frac{y}{z} - \left(y - b\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{a - t}{\color{blue}{\frac{y}{z} - \left(y - b\right)}} \]

      18. lower--.f64 82.9

          \[\leadsto x \cdot \frac{-1}{z - 1} - \frac{a - t}{\frac{y}{z} - \color{blue}{\left(y - b\right)}} \]

   10. Applied rewrites 82.9%

       \[\leadsto \color{blue}{x \cdot \frac{-1}{z - 1} - \frac{a - t}{\frac{y}{z} - \left(y - b\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-291}:\\ \;\;\;\;\frac{t\_1}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 10^{+286}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- t a) z (* y x)))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma (/ y (fma z (- b y) y)) x (/ (- t a) (- b y)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -4e-291)
       (/ t_1 t_2)
       (if (<= t_3 0.0)
         t_4
         (if (<= t_3 1e+286)
           (/ 1.0 (/ t_2 t_1))
           (fma (/ -1.0 (- z 1.0)) x (/ (- t a) (+ (- b y) (/ y z))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t - a), z, (y * x));
	double t_2 = fma((b - y), z, y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma((y / fma(z, (b - y), y)), x, ((t - a) / (b - y)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -4e-291) {
		tmp = t_1 / t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 1e+286) {
		tmp = 1.0 / (t_2 / t_1);
	} else {
		tmp = fma((-1.0 / (z - 1.0)), x, ((t - a) / ((b - y) + (y / z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(t - a), z, Float64(y * x))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(Float64(y / fma(z, Float64(b - y), y)), x, Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -4e-291)
		tmp = Float64(t_1 / t_2);
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 1e+286)
		tmp = Float64(1.0 / Float64(t_2 / t_1));
	else
		tmp = fma(Float64(-1.0 / Float64(z - 1.0)), x, Float64(Float64(t - a) / Float64(Float64(b - y) + Float64(y / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -4e-291], N[(t$95$1 / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 1e+286], N[(1.0 / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or -3.99999999999999985e-291 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 76.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   8. Applied rewrites 76.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -3.99999999999999985e-291

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right) + x \cdot y}}{y + z \cdot \left(b - y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right)} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - a\right) \cdot z} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      5. lower-fma.f64 67.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}}{y + z \cdot \left(b - y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}{y + z \cdot \left(b - y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-*.f64 67.3

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{y + z \cdot \left(b - y\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right) + y}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right)} + y} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\left(b - y\right) \cdot z} + y} \]

      13. lower-fma.f64 67.3

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   3. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   if -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000003e286

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      4. lower-/.f64 67.2

         \[\leadsto \frac{1}{\color{blue}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y + z \cdot \left(b - y\right)}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{z \cdot \left(b - y\right) + y}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{z \cdot \left(b - y\right)} + y}{x \cdot y + z \cdot \left(t - a\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - y\right) \cdot z} + y}{x \cdot y + z \cdot \left(t - a\right)}} \]

      9. lower-fma.f64 67.2

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{z \cdot \left(t - a\right) + x \cdot y}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{z \cdot \left(t - a\right)} + x \cdot y}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{\left(t - a\right) \cdot z} + x \cdot y}} \]

      14. lower-fma.f64 67.2

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}} \]

      17. lower-*.f64 67.2

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}} \]

   3. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, y \cdot x\right)}}} \]

   if 1.00000000000000003e286 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

      2. lower--.f64 82.9

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z - \color{blue}{1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_5 := \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, t\_3\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-291}:\\ \;\;\;\;\frac{t\_1}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 10^{+286}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- t a) z (* y x)))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (- t a) (- b y)))
        (t_4 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_5 (fma (/ y (fma z (- b y) y)) x t_3)))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -4e-291)
       (/ t_1 t_2)
       (if (<= t_4 0.0)
         t_5
         (if (<= t_4 1e+286)
           (/ 1.0 (/ t_2 t_1))
           (fma (/ -1.0 (- z 1.0)) x t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t - a), z, (y * x));
	double t_2 = fma((b - y), z, y);
	double t_3 = (t - a) / (b - y);
	double t_4 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_5 = fma((y / fma(z, (b - y), y)), x, t_3);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -4e-291) {
		tmp = t_1 / t_2;
	} else if (t_4 <= 0.0) {
		tmp = t_5;
	} else if (t_4 <= 1e+286) {
		tmp = 1.0 / (t_2 / t_1);
	} else {
		tmp = fma((-1.0 / (z - 1.0)), x, t_3);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(t - a), z, Float64(y * x))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	t_4 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_5 = fma(Float64(y / fma(z, Float64(b - y), y)), x, t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -4e-291)
		tmp = Float64(t_1 / t_2);
	elseif (t_4 <= 0.0)
		tmp = t_5;
	elseif (t_4 <= 1e+286)
		tmp = Float64(1.0 / Float64(t_2 / t_1));
	else
		tmp = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_3);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -4e-291], N[(t$95$1 / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.0], t$95$5, If[LessEqual[t$95$4, 1e+286], N[(1.0 / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or -3.99999999999999985e-291 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 76.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   8. Applied rewrites 76.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -3.99999999999999985e-291

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right) + x \cdot y}}{y + z \cdot \left(b - y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right)} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - a\right) \cdot z} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      5. lower-fma.f64 67.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}}{y + z \cdot \left(b - y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}{y + z \cdot \left(b - y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-*.f64 67.3

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{y + z \cdot \left(b - y\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right) + y}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right)} + y} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\left(b - y\right) \cdot z} + y} \]

      13. lower-fma.f64 67.3

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   3. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   if -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000003e286

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      4. lower-/.f64 67.2

         \[\leadsto \frac{1}{\color{blue}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y + z \cdot \left(b - y\right)}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{z \cdot \left(b - y\right) + y}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{z \cdot \left(b - y\right)} + y}{x \cdot y + z \cdot \left(t - a\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - y\right) \cdot z} + y}{x \cdot y + z \cdot \left(t - a\right)}} \]

      9. lower-fma.f64 67.2

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}}{x \cdot y + z \cdot \left(t - a\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{z \cdot \left(t - a\right) + x \cdot y}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{z \cdot \left(t - a\right)} + x \cdot y}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{\left(t - a\right) \cdot z} + x \cdot y}} \]

      14. lower-fma.f64 67.2

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\color{blue}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}} \]

      17. lower-*.f64 67.2

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}} \]

   3. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(b - y, z, y\right)}{\mathsf{fma}\left(t - a, z, y \cdot x\right)}}} \]

   if 1.00000000000000003e286 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

      2. lower--.f64 82.9

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z - \color{blue}{1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   10. Step-by-step derivation

       1. lower--.f64 65.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, t\_1\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-291}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 10^{+286}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (fma (- t a) z (* y x)) (fma (- b y) z y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma (/ y (fma z (- b y) y)) x t_1)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -4e-291)
       t_2
       (if (<= t_3 0.0)
         t_4
         (if (<= t_3 1e+286) t_2 (fma (/ -1.0 (- z 1.0)) x t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma((y / fma(z, (b - y), y)), x, t_1);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -4e-291) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 1e+286) {
		tmp = t_2;
	} else {
		tmp = fma((-1.0 / (z - 1.0)), x, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(Float64(y / fma(z, Float64(b - y), y)), x, t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -4e-291)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 1e+286)
		tmp = t_2;
	else
		tmp = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -4e-291], t$95$2, If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 1e+286], t$95$2, N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or -3.99999999999999985e-291 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 76.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   8. Applied rewrites 76.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -3.99999999999999985e-291 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000003e286

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right) + x \cdot y}}{y + z \cdot \left(b - y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right)} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - a\right) \cdot z} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      5. lower-fma.f64 67.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}}{y + z \cdot \left(b - y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}{y + z \cdot \left(b - y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-*.f64 67.3

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{y + z \cdot \left(b - y\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right) + y}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right)} + y} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\left(b - y\right) \cdot z} + y} \]

      13. lower-fma.f64 67.3

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   3. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   if 1.00000000000000003e286 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

      2. lower--.f64 82.9

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z - \color{blue}{1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   10. Step-by-step derivation

       1. lower--.f64 65.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_2\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-291}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+286}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (- t a) z (* y x)) (fma (- b y) z y)))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma (/ -1.0 (- z 1.0)) x t_2)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -4e-291)
       t_1
       (if (<= t_3 0.0) t_2 (if (<= t_3 1e+286) t_1 t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma((-1.0 / (z - 1.0)), x, t_2);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -4e-291) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 1e+286) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -4e-291)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 1e+286)
		tmp = t_1;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -4e-291], t$95$1, If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 1e+286], t$95$1, t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.00000000000000003e286 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

      2. lower--.f64 82.9

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z - \color{blue}{1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   10. Step-by-step derivation

       1. lower--.f64 65.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -3.99999999999999985e-291 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000003e286

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right) + x \cdot y}}{y + z \cdot \left(b - y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(t - a\right)} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - a\right) \cdot z} + x \cdot y}{y + z \cdot \left(b - y\right)} \]

      5. lower-fma.f64 67.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t - a, z, x \cdot y\right)}}{y + z \cdot \left(b - y\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{x \cdot y}\right)}{y + z \cdot \left(b - y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-*.f64 67.3

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, \color{blue}{y \cdot x}\right)}{y + z \cdot \left(b - y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{y + z \cdot \left(b - y\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right) + y}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{z \cdot \left(b - y\right)} + y} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\left(b - y\right) \cdot z} + y} \]

      13. lower-fma.f64 67.3

          \[\leadsto \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   3. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   if -3.99999999999999985e-291 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.0

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \frac{t - a}{b - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) INFINITY)
   (fma (/ y (fma z (- b y) y)) x (/ (- t a) (+ (- b y) (/ y z))))
   (fma (/ x (fma (- b y) (/ z y) 1.0)) (/ y y) (/ (- t a) (- b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= ((double) INFINITY)) {
		tmp = fma((y / fma(z, (b - y), y)), x, ((t - a) / ((b - y) + (y / z))));
	} else {
		tmp = fma((x / fma((b - y), (z / y), 1.0)), (y / y), ((t - a) / (b - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y)))) <= Inf)
		tmp = fma(Float64(y / fma(z, Float64(b - y), y)), x, Float64(Float64(t - a) / Float64(Float64(b - y) + Float64(y / z))));
	else
		tmp = fma(Float64(x / fma(Float64(b - y), Float64(z / y), 1.0)), Float64(y / y), Float64(Float64(t - a) / Float64(b - y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(b - y), $MachinePrecision] * N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / y), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \color{blue}{\frac{t - a}{b - y}}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \frac{t - a}{\color{blue}{b - y}}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \frac{t - a}{\color{blue}{b} - y}\right) \]

      3. lower--.f64 80.2

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \frac{t - a}{b - \color{blue}{y}}\right) \]

   6. Applied rewrites 80.2%

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \color{blue}{\frac{t - a}{b - y}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (/ y (fma z (- b y) y)) x (/ (- t a) (+ (- b y) (/ y z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((y / fma(z, (b - y), y)), x, ((t - a) / ((b - y) + (y / z))));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(y / fma(z, Float64(b - y), y)), x, Float64(Float64(t - a) / Float64(Float64(b - y) + Float64(y / z))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

   3. div-add N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   6. sum-to-mult N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   7. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

3. Applied rewrites 83.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   2. *-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   7. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   8. *-inverses N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   11. *-lft-identity N/A

       \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   13. *-inverses N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   14. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   15. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   16. associate-/l* N/A

       \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   17. *-commutative N/A

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

5. Applied rewrites 93.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]
6. Add Preprocessing

Alternative 8: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2900000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ -1.0 (- z 1.0)) x (/ (- t a) (- b y)))))
   (if (<= z -2.7e-16)
     t_1
     (if (<= z 2900000000000.0)
       (/ (fma y x (* (- t a) z)) (+ y (* z b)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((-1.0 / (z - 1.0)), x, ((t - a) / (b - y)));
	double tmp;
	if (z <= -2.7e-16) {
		tmp = t_1;
	} else if (z <= 2900000000000.0) {
		tmp = fma(y, x, ((t - a) * z)) / (y + (z * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(-1.0 / Float64(z - 1.0)), x, Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (z <= -2.7e-16)
		tmp = t_1;
	elseif (z <= 2900000000000.0)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(y + Float64(z * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.7e-16], t$95$1, If[LessEqual[z, 2900000000000.0], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.69999999999999999e-16 or 2.9e12 < z

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

      2. lower--.f64 82.9

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z - \color{blue}{1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   10. Step-by-step derivation

       1. lower--.f64 65.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]

   if -2.69999999999999999e-16 < z < 2.9e12

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x} + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      4. lower-fma.f64 67.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, z \cdot \left(t - a\right)\right)}}{y + z \cdot \left(b - y\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(t - a\right)}\right)}{y + z \cdot \left(b - y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(t - a\right) \cdot z}\right)}{y + z \cdot \left(b - y\right)} \]

      7. lower-*.f64 67.3

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(t - a\right) \cdot z}\right)}{y + z \cdot \left(b - y\right)} \]

   3. Applied rewrites 67.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}}{y + z \cdot \left(b - y\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \color{blue}{b}} \]
   5. Step-by-step derivation

   6. Applied rewrites 57.7%

      \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right)\\ \mathbf{elif}\;z \leq 0.0098:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ -1.0 (- z 1.0)) x (/ (- t a) (- b y)))))
   (if (<= z -3.5e-73)
     t_1
     (if (<= z 1e-82)
       (fma 1.0 x (/ (* z (- t a)) y))
       (if (<= z 0.0098) (/ (fma t z (* x y)) (+ y (* z (- b y)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((-1.0 / (z - 1.0)), x, ((t - a) / (b - y)));
	double tmp;
	if (z <= -3.5e-73) {
		tmp = t_1;
	} else if (z <= 1e-82) {
		tmp = fma(1.0, x, ((z * (t - a)) / y));
	} else if (z <= 0.0098) {
		tmp = fma(t, z, (x * y)) / (y + (z * (b - y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(-1.0 / Float64(z - 1.0)), x, Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (z <= -3.5e-73)
		tmp = t_1;
	elseif (z <= 1e-82)
		tmp = fma(1.0, x, Float64(Float64(z * Float64(t - a)) / y));
	elseif (z <= 0.0098)
		tmp = Float64(fma(t, z, Float64(x * y)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e-73], t$95$1, If[LessEqual[z, 1e-82], N[(1.0 * x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0098], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4999999999999998e-73 or 0.0097999999999999997 < z

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

      2. lower--.f64 82.9

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z - \color{blue}{1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   10. Step-by-step derivation

       1. lower--.f64 65.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]

   if -3.4999999999999998e-73 < z < 1e-82

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{z \cdot \left(t - a\right)}{y}}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{\color{blue}{y}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right) \]

       3. lower--.f64 38.7

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right) \]

   11. Applied rewrites 38.7%

       \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{z \cdot \left(t - a\right)}{y}}\right) \]

   if 1e-82 < z < 0.0097999999999999997

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   2. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{t \cdot z + x \cdot y}}{y + z \cdot \left(b - y\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right)}{y + z \cdot \left(b - y\right)} \]

      2. lower-*.f64 47.9

         \[\leadsto \frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{y + z \cdot \left(b - y\right)} \]

   4. Applied rewrites 47.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}}{y + z \cdot \left(b - y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ -1.0 (- z 1.0)) x (/ (- t a) (- b y)))))
   (if (<= z -3.5e-73)
     t_1
     (if (<= z 1.2e-69) (fma 1.0 x (/ (* z (- t a)) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((-1.0 / (z - 1.0)), x, ((t - a) / (b - y)));
	double tmp;
	if (z <= -3.5e-73) {
		tmp = t_1;
	} else if (z <= 1.2e-69) {
		tmp = fma(1.0, x, ((z * (t - a)) / y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(-1.0 / Float64(z - 1.0)), x, Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (z <= -3.5e-73)
		tmp = t_1;
	elseif (z <= 1.2e-69)
		tmp = fma(1.0, x, Float64(Float64(z * Float64(t - a)) / y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e-73], t$95$1, If[LessEqual[z, 1.2e-69], N[(1.0 * x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4999999999999998e-73 or 1.2000000000000001e-69 < z

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

      2. lower--.f64 82.9

         \[\leadsto \mathsf{fma}\left(\frac{-1}{z - \color{blue}{1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{z - 1}}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]
   10. Step-by-step derivation

       1. lower--.f64 65.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - \color{blue}{y}}\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{\color{blue}{b - y}}\right) \]

   if -3.4999999999999998e-73 < z < 1.2000000000000001e-69

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{z \cdot \left(t - a\right)}{y}}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{\color{blue}{y}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right) \]

       3. lower--.f64 38.7

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right) \]

   11. Applied rewrites 38.7%

       \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{z \cdot \left(t - a\right)}{y}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(t - a\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{t\_1}{y}\right)\\ \mathbf{elif}\;z \leq 22000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t\_1\right)}{b \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- t a))) (t_2 (/ (- t a) (- b y))))
   (if (<= z -3.2e-15)
     t_2
     (if (<= z 1.15e-82)
       (fma 1.0 x (/ t_1 y))
       (if (<= z 22000000000000.0) (/ (fma x y t_1) (* b z)) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (t - a);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.2e-15) {
		tmp = t_2;
	} else if (z <= 1.15e-82) {
		tmp = fma(1.0, x, (t_1 / y));
	} else if (z <= 22000000000000.0) {
		tmp = fma(x, y, t_1) / (b * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(t - a))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.2e-15)
		tmp = t_2;
	elseif (z <= 1.15e-82)
		tmp = fma(1.0, x, Float64(t_1 / y));
	elseif (z <= 22000000000000.0)
		tmp = Float64(fma(x, y, t_1) / Float64(b * z));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e-15], t$95$2, If[LessEqual[z, 1.15e-82], N[(1.0 * x + N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 22000000000000.0], N[(N[(x * y + t$95$1), $MachinePrecision] / N[(b * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1999999999999999e-15 or 2.2e13 < z

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.0

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.1999999999999999e-15 < z < 1.14999999999999998e-82

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{z \cdot \left(t - a\right)}{y}}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{\color{blue}{y}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right) \]

       3. lower--.f64 38.7

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right) \]

   11. Applied rewrites 38.7%

       \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{z \cdot \left(t - a\right)}{y}}\right) \]

   if 1.14999999999999998e-82 < z < 2.2e13

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{b \cdot z}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y + z \cdot \left(t - a\right)}{\color{blue}{b \cdot z}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\color{blue}{b} \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{b \cdot z} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{b \cdot z} \]

      5. lower-*.f64 28.8

         \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{b \cdot \color{blue}{z}} \]

   4. Applied rewrites 28.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{b \cdot z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.2e-15)
     t_1
     (if (<= z 9.8e-37) (fma 1.0 x (/ (* z (- t a)) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.2e-15) {
		tmp = t_1;
	} else if (z <= 9.8e-37) {
		tmp = fma(1.0, x, ((z * (t - a)) / y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.2e-15)
		tmp = t_1;
	elseif (z <= 9.8e-37)
		tmp = fma(1.0, x, Float64(Float64(z * Float64(t - a)) / y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e-15], t$95$1, If[LessEqual[z, 9.8e-37], N[(1.0 * x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999999e-15 or 9.80000000000000036e-37 < z

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.0

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.1999999999999999e-15 < z < 9.80000000000000036e-37

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{z \cdot \left(t - a\right)}{y}}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{\color{blue}{y}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right) \]

       3. lower--.f64 38.7

          \[\leadsto \mathsf{fma}\left(1, x, \frac{z \cdot \left(t - a\right)}{y}\right) \]

   11. Applied rewrites 38.7%

       \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{z \cdot \left(t - a\right)}{y}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-40}:\\ \;\;\;\;-1 \cdot \frac{x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.1e+22) t_1 (if (<= z 3.4e-40) (* -1.0 (/ x (- z 1.0))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.1e+22) {
		tmp = t_1;
	} else if (z <= 3.4e-40) {
		tmp = -1.0 * (x / (z - 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-3.1d+22)) then
        tmp = t_1
    else if (z <= 3.4d-40) then
        tmp = (-1.0d0) * (x / (z - 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.1e+22) {
		tmp = t_1;
	} else if (z <= 3.4e-40) {
		tmp = -1.0 * (x / (z - 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -3.1e+22:
		tmp = t_1
	elif z <= 3.4e-40:
		tmp = -1.0 * (x / (z - 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.1e+22)
		tmp = t_1;
	elseif (z <= 3.4e-40)
		tmp = Float64(-1.0 * Float64(x / Float64(z - 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -3.1e+22)
		tmp = t_1;
	elseif (z <= 3.4e-40)
		tmp = -1.0 * (x / (z - 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.1e+22], t$95$1, If[LessEqual[z, 3.4e-40], N[(-1.0 * N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1000000000000002e22 or 3.39999999999999984e-40 < z

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

      3. lower--.f64 51.0

         \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -3.1000000000000002e22 < z < 3.39999999999999984e-40

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   2. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{z - 1}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{z - 1}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x}{\color{blue}{z - 1}} \]

      3. lower--.f64 33.2

         \[\leadsto -1 \cdot \frac{x}{z - \color{blue}{1}} \]

   4. Applied rewrites 33.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{z - 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 51.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{t - a}{b - y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ (- t a) (- b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (t - a) / (b - y);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (t - a) / (b - y)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (t - a) / (b - y);
}

Python

def code(x, y, z, t, a, b):
	return (t - a) / (b - y)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(t - a) / Float64(b - y))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (t - a) / (b - y);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

2. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{t - a}{\color{blue}{b} - y} \]

   3. lower--.f64 51.0

      \[\leadsto \frac{t - a}{b - \color{blue}{y}} \]

4. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
5. Add Preprocessing

Alternative 15: 37.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b}\\ \mathbf{if}\;b \leq -67000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 10^{-63}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) b)))
   (if (<= b -67000000000000.0) t_1 (if (<= b 1e-63) (/ t (- b y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (b <= -67000000000000.0) {
		tmp = t_1;
	} else if (b <= 1e-63) {
		tmp = t / (b - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / b
    if (b <= (-67000000000000.0d0)) then
        tmp = t_1
    else if (b <= 1d-63) then
        tmp = t / (b - y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (b <= -67000000000000.0) {
		tmp = t_1;
	} else if (b <= 1e-63) {
		tmp = t / (b - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / b
	tmp = 0
	if b <= -67000000000000.0:
		tmp = t_1
	elif b <= 1e-63:
		tmp = t / (b - y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / b)
	tmp = 0.0
	if (b <= -67000000000000.0)
		tmp = t_1;
	elseif (b <= 1e-63)
		tmp = Float64(t / Float64(b - y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / b;
	tmp = 0.0;
	if (b <= -67000000000000.0)
		tmp = t_1;
	elseif (b <= 1e-63)
		tmp = t / (b - y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, -67000000000000.0], t$95$1, If[LessEqual[b, 1e-63], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.7e13 or 1.00000000000000007e-63 < b

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b}} \]

      2. lower--.f64 35.2

         \[\leadsto \frac{t - a}{b} \]

   4. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]

   if -6.7e13 < b < 1.00000000000000007e-63

   1. Initial program 67.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      6. sum-to-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      13. *-inverses N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      15. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      16. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{t}{\left(b + \frac{y}{z}\right) - y}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t}{\color{blue}{\left(b + \frac{y}{z}\right) - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{t}{\left(b + \frac{y}{z}\right) - \color{blue}{y}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{t}{\left(b + \frac{y}{z}\right) - y} \]

      4. lower-/.f64 32.4

         \[\leadsto \frac{t}{\left(b + \frac{y}{z}\right) - y} \]

   8. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\frac{t}{\left(b + \frac{y}{z}\right) - y}} \]

   9. Taylor expanded in z around inf

      \[\leadsto \frac{t}{\color{blue}{b - y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{t}{b - \color{blue}{y}} \]

       2. lower--.f64 27.4

          \[\leadsto \frac{t}{b - y} \]

   11. Applied rewrites 27.4%

       \[\leadsto \frac{t}{\color{blue}{b - y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 27.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{t}{b - y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ t (- b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return t / (b - y);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = t / (b - y)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return t / (b - y);
}

Python

def code(x, y, z, t, a, b):
	return t / (b - y)

Julia

function code(x, y, z, t, a, b)
	return Float64(t / Float64(b - y))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = t / (b - y);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y + z \cdot \left(t - a\right)}}{y + z \cdot \left(b - y\right)} \]

   3. div-add N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{y + z \cdot \left(b - y\right)}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   6. sum-to-mult N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(1 + \frac{z \cdot \left(b - y\right)}{y}\right) \cdot y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   7. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}} \cdot \frac{y}{y}} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{1 + \frac{z \cdot \left(b - y\right)}{y}}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z \cdot \left(b - y\right)}{y} + 1}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{z \cdot \left(b - y\right)}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{\color{blue}{\left(b - y\right) \cdot z}}{y} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\left(b - y\right) \cdot \frac{z}{y}} + 1}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \color{blue}{\frac{z}{y}}, 1\right)}, \frac{y}{y}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \color{blue}{\frac{y}{y}}, \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\right) \]

3. Applied rewrites 83.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   2. *-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x \cdot 1}}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot 1}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot \frac{y}{y}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}}, \frac{y}{y}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   7. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{\frac{y}{y}}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   8. *-inverses N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}, \color{blue}{1}, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right) \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}\right)} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   11. *-lft-identity N/A

       \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{\frac{y}{y}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   13. *-inverses N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot \color{blue}{1} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   14. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} \cdot 1 + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   15. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{x \cdot 1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   16. associate-/l* N/A

       \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)}} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

   17. *-commutative N/A

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b - y, \frac{z}{y}, 1\right)} \cdot x} + \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \]

5. Applied rewrites 93.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, x, \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\right)} \]

6. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{t}{\left(b + \frac{y}{z}\right) - y}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{t}{\color{blue}{\left(b + \frac{y}{z}\right) - y}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{t}{\left(b + \frac{y}{z}\right) - \color{blue}{y}} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{t}{\left(b + \frac{y}{z}\right) - y} \]

   4. lower-/.f64 32.4

      \[\leadsto \frac{t}{\left(b + \frac{y}{z}\right) - y} \]

8. Applied rewrites 32.4%

   \[\leadsto \color{blue}{\frac{t}{\left(b + \frac{y}{z}\right) - y}} \]

9. Taylor expanded in z around inf

   \[\leadsto \frac{t}{\color{blue}{b - y}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{t}{b - \color{blue}{y}} \]

    2. lower--.f64 27.4

       \[\leadsto \frac{t}{b - y} \]

11. Applied rewrites 27.4%

    \[\leadsto \frac{t}{\color{blue}{b - y}} \]
12. Add Preprocessing

Alternative 17: 19.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{t}{b} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ t b))

C

double code(double x, double y, double z, double t, double a, double b) {
	return t / b;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = t / b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return t / b;
}

Python

def code(x, y, z, t, a, b):
	return t / b

Julia

function code(x, y, z, t, a, b)
	return Float64(t / b)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = t / b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(t / b), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{t - a}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{t - a}{\color{blue}{b}} \]

   2. lower--.f64 35.2

      \[\leadsto \frac{t - a}{b} \]

4. Applied rewrites 35.2%

   \[\leadsto \color{blue}{\frac{t - a}{b}} \]

5. Taylor expanded in t around inf

   \[\leadsto \frac{t}{\color{blue}{b}} \]
6. Step-by-step derivation

   1. lower-/.f64 19.8

      \[\leadsto \frac{t}{b} \]

7. Applied rewrites 19.8%

   \[\leadsto \frac{t}{\color{blue}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64
  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))