Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.4% → 95.4%

Time: 5.2s

Alternatives: 17

Speedup: 1.1×

Specification

Math

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

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: 66.4% accurate, 1.0× speedup

Math

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

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: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_3 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{x \cdot y}{b - y}, \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}\right)}{z}, \frac{t}{b - y}\right) - \frac{a}{b - y}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_3 (fma (/ -1.0 (- z 1.0)) x (/ (- t a) (- b y)))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e-248)
       (/ (fma (- t a) z (* y x)) (fma (- b y) z y))
       (if (<= t_2 0.0)
         (-
          (fma
           -1.0
           (/
            (fma -1.0 (/ (* x y) (- b y)) (/ (* y (- t a)) (pow (- b y) 2.0)))
            z)
           (/ t (- b y)))
          (/ a (- b y)))
         (if (<= t_2 5e+283) (/ (fma y x (* (- t a) z)) t_1) t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = ((x * y) + (z * (t - a))) / t_1;
	double t_3 = fma((-1.0 / (z - 1.0)), x, ((t - a) / (b - y)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e-248) {
		tmp = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	} else if (t_2 <= 0.0) {
		tmp = fma(-1.0, (fma(-1.0, ((x * y) / (b - y)), ((y * (t - a)) / pow((b - y), 2.0))) / z), (t / (b - y))) - (a / (b - y));
	} else if (t_2 <= 5e+283) {
		tmp = fma(y, x, ((t - a) * z)) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_3 = fma(Float64(-1.0 / Float64(z - 1.0)), x, Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e-248)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y));
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(-1.0, Float64(fma(-1.0, Float64(Float64(x * y) / Float64(b - y)), Float64(Float64(y * Float64(t - a)) / (Float64(b - y) ^ 2.0))) / z), Float64(t / Float64(b - y))) - Float64(a / Float64(b - y)));
	elseif (t_2 <= 5e+283)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e-248], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(-1.0 * N[(N[(-1.0 * N[(N[(x * y), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+283], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]]

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 5.0000000000000004e283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999998e-249

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 66.4

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

      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 66.4

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

      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 66.4

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

   3. Applied rewrites 66.4%

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

   if -9.9999999999999998e-249 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} + \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \frac{t}{b - y}\right) - \frac{a}{b - y}} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} + \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \frac{t}{b - y}\right) - \color{blue}{\frac{a}{b - y}} \]

   6. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{x \cdot y}{b - y}, \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}\right)}{z}, \frac{t}{b - y}\right) - \frac{a}{b - y}} \]

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000004e283

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 66.4

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

      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 66.4

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

   3. Applied rewrites 66.4%

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

Alternative 2: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_5 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_3\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t\_2}, x, t\_3\right)\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (- t a) (- b y)))
        (t_4 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_5 (fma (/ -1.0 (- z 1.0)) x t_3)))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -1e-248)
       (/ (fma (- t a) z (* y x)) t_2)
       (if (<= t_4 2e-286)
         (fma (/ y t_2) x t_3)
         (if (<= t_4 5e+283) (/ (fma y x (* (- t a) z)) t_1) t_5))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = fma((b - y), z, y);
	double t_3 = (t - a) / (b - y);
	double t_4 = ((x * y) + (z * (t - a))) / t_1;
	double t_5 = fma((-1.0 / (z - 1.0)), x, t_3);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -1e-248) {
		tmp = fma((t - a), z, (y * x)) / t_2;
	} else if (t_4 <= 2e-286) {
		tmp = fma((y / t_2), x, t_3);
	} else if (t_4 <= 5e+283) {
		tmp = fma(y, x, ((t - a) * z)) / t_1;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	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))) / t_1)
	t_5 = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -1e-248)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / t_2);
	elseif (t_4 <= 2e-286)
		tmp = fma(Float64(y / t_2), x, t_3);
	elseif (t_4 <= 5e+283)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1);
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $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] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -1e-248], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 2e-286], N[(N[(y / t$95$2), $MachinePrecision] * x + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 5e+283], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$5]]]]]]]]]

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 5.0000000000000004e283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999998e-249

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 66.4

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

      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 66.4

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

      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 66.4

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

   3. Applied rewrites 66.4%

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

   if -9.9999999999999998e-249 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.0000000000000001e-286

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   if 2.0000000000000001e-286 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000004e283

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 66.4

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

      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 66.4

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

   3. Applied rewrites 66.4%

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

Alternative 3: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_2\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (fma (/ -1.0 (- z 1.0)) x t_2)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-248)
       (/ (fma (- t a) z (* y x)) (fma (- b y) z y))
       (if (<= t_3 0.0)
         t_2
         (if (<= t_3 5e+283) (/ (fma y x (* (- t a) z)) t_1) t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = (t - a) / (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	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 <= -1e-248) {
		tmp = fma((t - a), z, (y * x)) / fma((b - y), z, y);
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 5e+283) {
		tmp = fma(y, x, ((t - a) * z)) / t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	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 <= -1e-248)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / fma(Float64(b - y), z, y));
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 5e+283)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $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] / t$95$1), $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, -1e-248], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 5e+283], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$4]]]]]]]]

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 5.0000000000000004e283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999998e-249

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 66.4

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

      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 66.4

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

      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 66.4

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

   3. Applied rewrites 66.4%

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

   if -9.9999999999999998e-249 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000004e283

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 66.4

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

      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 66.4

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

   3. Applied rewrites 66.4%

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

Alternative 4: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_2\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

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 -1e-248)
       t_1
       (if (<= t_3 0.0) t_2 (if (<= t_3 5e+283) 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 <= -1e-248) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 5e+283) {
		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 <= -1e-248)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 5e+283)
		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, -1e-248], t$95$1, If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 5e+283], 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 5.0000000000000004e283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999998e-249 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000004e283

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 66.4

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

      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 66.4

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

      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 66.4

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

   3. Applied rewrites 66.4%

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

   if -9.9999999999999998e-249 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

Alternative 5: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_2\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+177}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{\mathsf{fma}\left(b, z, y\right)}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (fma (/ -1.0 (- z 1.0)) x t_2)))
   (if (<= t_1 -5e+177)
     t_3
     (if (<= t_1 -1e-248)
       (/ (fma (- t a) z (* x y)) (fma b z y))
       (if (<= t_1 0.0)
         t_2
         (if (<= t_1 5e+283)
           (/ (fma y x (* (- t a) z)) (+ y (* z b)))
           t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = (t - a) / (b - y);
	double t_3 = fma((-1.0 / (z - 1.0)), x, t_2);
	double tmp;
	if (t_1 <= -5e+177) {
		tmp = t_3;
	} else if (t_1 <= -1e-248) {
		tmp = fma((t - a), z, (x * y)) / fma(b, z, y);
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+283) {
		tmp = fma(y, x, ((t - a) * z)) / (y + (z * b));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_2)
	tmp = 0.0
	if (t_1 <= -5e+177)
		tmp = t_3;
	elseif (t_1 <= -1e-248)
		tmp = Float64(fma(Float64(t - a), z, Float64(x * y)) / fma(b, z, y));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+283)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(y + Float64(z * b)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+177], t$95$3, If[LessEqual[t$95$1, -1e-248], N[(N[(N[(t - a), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(b * z + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 5e+283], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

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)))) < -5.0000000000000003e177 or 5.0000000000000004e283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   if -5.0000000000000003e177 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999998e-249

   1. Initial program 66.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 57.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 57.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if -9.9999999999999998e-249 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000004e283

   1. Initial program 66.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 57.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 57.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

Alternative 6: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{\mathsf{fma}\left(b, z, y\right)}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \mathsf{fma}\left(\frac{-1}{z - 1}, x, t\_3\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+177}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (- t a) z (* x y)) (fma b z y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (/ (- t a) (- b y)))
        (t_4 (fma (/ -1.0 (- z 1.0)) x t_3)))
   (if (<= t_2 -5e+177)
     t_4
     (if (<= t_2 -1e-248)
       t_1
       (if (<= t_2 0.0) t_3 (if (<= t_2 5e+283) 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, (x * y)) / fma(b, z, y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = (t - a) / (b - y);
	double t_4 = fma((-1.0 / (z - 1.0)), x, t_3);
	double tmp;
	if (t_2 <= -5e+177) {
		tmp = t_4;
	} else if (t_2 <= -1e-248) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 5e+283) {
		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(x * y)) / fma(b, z, y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	t_4 = fma(Float64(-1.0 / Float64(z - 1.0)), x, t_3)
	tmp = 0.0
	if (t_2 <= -5e+177)
		tmp = t_4;
	elseif (t_2 <= -1e-248)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 5e+283)
		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[(x * y), $MachinePrecision]), $MachinePrecision] / N[(b * z + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+177], t$95$4, If[LessEqual[t$95$2, -1e-248], t$95$1, If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 5e+283], 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)))) < -5.0000000000000003e177 or 5.0000000000000004e283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   if -5.0000000000000003e177 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999998e-249 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000004e283

   1. Initial program 66.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 57.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 57.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if -9.9999999999999998e-249 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

Alternative 7: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -10500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, x \cdot y\right)}{\mathsf{fma}\left(b, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, x, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -10500.0)
     t_1
     (if (<= z 1.2)
       (/ (fma (- t a) z (* x y)) (fma b z y))
       (fma (/ -1.0 z) 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 tmp;
	if (z <= -10500.0) {
		tmp = t_1;
	} else if (z <= 1.2) {
		tmp = fma((t - a), z, (x * y)) / fma(b, z, y);
	} else {
		tmp = fma((-1.0 / z), x, 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 <= -10500.0)
		tmp = t_1;
	elseif (z <= 1.2)
		tmp = Float64(fma(Float64(t - a), z, Float64(x * y)) / fma(b, z, y));
	else
		tmp = fma(Float64(-1.0 / z), 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]}, If[LessEqual[z, -10500.0], t$95$1, If[LessEqual[z, 1.2], N[(N[(N[(t - a), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(b * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * x + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -10500

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if -10500 < z < 1.19999999999999996

   1. Initial program 66.4%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 57.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 57.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 1.19999999999999996 < z

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   10. Taylor expanded in z around inf

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

       1. lower-/.f64 47.6

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

   12. Applied rewrites 47.6%

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

Alternative 8: 74.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{t\_1}\\ t_3 := \mathsf{fma}\left(1, x, \frac{-1}{y - b} \cdot \left(t - a\right)\right)\\ t_4 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_5 := \frac{t - a}{b - y}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+169}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{t\_1}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, x, t\_5\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (/ (fma t z (* x y)) t_1))
        (t_3 (fma 1.0 x (* (/ -1.0 (- y b)) (- t a))))
        (t_4 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_5 (/ (- t a) (- b y))))
   (if (<= t_4 -1e+169)
     t_3
     (if (<= t_4 -1e+126)
       (/ (* (- t a) z) t_1)
       (if (<= t_4 -1e-248)
         t_2
         (if (<= t_4 0.0)
           t_5
           (if (<= t_4 5e+283)
             t_2
             (if (<= t_4 INFINITY) t_3 (fma (/ -1.0 z) x t_5)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = fma(t, z, (x * y)) / t_1;
	double t_3 = fma(1.0, x, ((-1.0 / (y - b)) * (t - a)));
	double t_4 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_5 = (t - a) / (b - y);
	double tmp;
	if (t_4 <= -1e+169) {
		tmp = t_3;
	} else if (t_4 <= -1e+126) {
		tmp = ((t - a) * z) / t_1;
	} else if (t_4 <= -1e-248) {
		tmp = t_2;
	} else if (t_4 <= 0.0) {
		tmp = t_5;
	} else if (t_4 <= 5e+283) {
		tmp = t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = fma((-1.0 / z), x, t_5);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(fma(t, z, Float64(x * y)) / t_1)
	t_3 = fma(1.0, x, Float64(Float64(-1.0 / Float64(y - b)) * Float64(t - a)))
	t_4 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_5 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (t_4 <= -1e+169)
		tmp = t_3;
	elseif (t_4 <= -1e+126)
		tmp = Float64(Float64(Float64(t - a) * z) / t_1);
	elseif (t_4 <= -1e-248)
		tmp = t_2;
	elseif (t_4 <= 0.0)
		tmp = t_5;
	elseif (t_4 <= 5e+283)
		tmp = t_2;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = fma(Float64(-1.0 / z), x, t_5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 * x + N[(N[(-1.0 / N[(y - b), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $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[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e+169], t$95$3, If[LessEqual[t$95$4, -1e+126], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, -1e-248], t$95$2, If[LessEqual[t$95$4, 0.0], t$95$5, If[LessEqual[t$95$4, 5e+283], t$95$2, If[LessEqual[t$95$4, Infinity], t$95$3, N[(N[(-1.0 / z), $MachinePrecision] * x + t$95$5), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999934e168 or 5.0000000000000004e283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower--.f64 76.5

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 48.7%

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

   if -9.99999999999999934e168 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999925e125

   1. Initial program 66.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 41.6

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

   4. Applied rewrites 41.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.6

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

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + z \cdot \left(b - y\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(+-commutative, \left(z \cdot \left(b - y\right) + y\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z \cdot \left(b - y\right)\right)\right) + y} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(lift-fma.f64, \left(\mathsf{fma}\left(z, b - y, y\right)\right)\right)} \]

   6. Applied rewrites 41.6%

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

   if -9.99999999999999925e125 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999998e-249 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000004e283

   1. Initial program 66.4%

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

   2. Taylor expanded in t around inf

      \[\leadsto \frac{\color{blue}{t \cdot z}}{y + z \cdot \left(b - y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 23.1

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

   4. Applied rewrites 23.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 23.1

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

   6. Applied rewrites 23.1%

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

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 47.4

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

   9. Applied rewrites 47.4%

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

   if -9.9999999999999998e-249 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

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

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   10. Taylor expanded in z around inf

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

       1. lower-/.f64 47.6

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

   12. Applied rewrites 47.6%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{z}}, x, \frac{t - a}{b - y}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{t - a}{y} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, x, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.85e+30)
     t_1
     (if (<= z -2.4e-72)
       (/ (* (- t a) z) (fma z (- b y) y))
       (if (<= z 0.095)
         (fma 1.0 x (* (/ (- t a) y) z))
         (fma (/ -1.0 z) 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 tmp;
	if (z <= -1.85e+30) {
		tmp = t_1;
	} else if (z <= -2.4e-72) {
		tmp = ((t - a) * z) / fma(z, (b - y), y);
	} else if (z <= 0.095) {
		tmp = fma(1.0, x, (((t - a) / y) * z));
	} else {
		tmp = fma((-1.0 / z), x, 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 <= -1.85e+30)
		tmp = t_1;
	elseif (z <= -2.4e-72)
		tmp = Float64(Float64(Float64(t - a) * z) / fma(z, Float64(b - y), y));
	elseif (z <= 0.095)
		tmp = fma(1.0, x, Float64(Float64(Float64(t - a) / y) * z));
	else
		tmp = fma(Float64(-1.0 / z), 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]}, If[LessEqual[z, -1.85e+30], t$95$1, If[LessEqual[z, -2.4e-72], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.095], N[(1.0 * x + N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * x + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.85000000000000008e30

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if -1.85000000000000008e30 < z < -2.4e-72

   1. Initial program 66.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 41.6

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

   4. Applied rewrites 41.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.6

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

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + z \cdot \left(b - y\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(+-commutative, \left(z \cdot \left(b - y\right) + y\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z \cdot \left(b - y\right)\right)\right) + y} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(lift-fma.f64, \left(\mathsf{fma}\left(z, b - y, y\right)\right)\right)} \]

   6. Applied rewrites 41.6%

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

   if -2.4e-72 < z < 0.095000000000000001

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 35.9

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

   9. Applied rewrites 35.9%

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

   if 0.095000000000000001 < z

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   6. Applied rewrites 76.6%

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

   7. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 66.2

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

   9. Applied rewrites 66.2%

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

   10. Taylor expanded in z around inf

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

       1. lower-/.f64 47.6

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

   12. Applied rewrites 47.6%

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

Alternative 10: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+30}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 0.054:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{t - a}{y} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.85e+30)
   (/ (- t a) (- b y))
   (if (<= z -2.4e-72)
     (/ (* (- t a) z) (fma z (- b y) y))
     (if (<= z 0.054)
       (fma 1.0 x (* (/ (- t a) y) z))
       (- (/ t (- b y)) (/ a (- b y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.85e+30) {
		tmp = (t - a) / (b - y);
	} else if (z <= -2.4e-72) {
		tmp = ((t - a) * z) / fma(z, (b - y), y);
	} else if (z <= 0.054) {
		tmp = fma(1.0, x, (((t - a) / y) * z));
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.85e+30)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= -2.4e-72)
		tmp = Float64(Float64(Float64(t - a) * z) / fma(z, Float64(b - y), y));
	elseif (z <= 0.054)
		tmp = fma(1.0, x, Float64(Float64(Float64(t - a) / y) * z));
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.85e+30], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.4e-72], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.054], N[(1.0 * x + N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.85000000000000008e30

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if -1.85000000000000008e30 < z < -2.4e-72

   1. Initial program 66.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 41.6

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

   4. Applied rewrites 41.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.6

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

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + z \cdot \left(b - y\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(+-commutative, \left(z \cdot \left(b - y\right) + y\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z \cdot \left(b - y\right)\right)\right) + y} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(lift-fma.f64, \left(\mathsf{fma}\left(z, b - y, y\right)\right)\right)} \]

   6. Applied rewrites 41.6%

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

   if -2.4e-72 < z < 0.0539999999999999994

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 35.9

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

   9. Applied rewrites 35.9%

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

   if 0.0539999999999999994 < z

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 50.9

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

   6. Applied rewrites 50.9%

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

Alternative 11: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 0.054:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{t - a}{y} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.95e-10)
   (/ (- t a) (- b y))
   (if (<= z 0.054)
     (fma 1.0 x (* (/ (- t a) y) z))
     (- (/ t (- b y)) (/ a (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.95e-10) {
		tmp = (t - a) / (b - y);
	} else if (z <= 0.054) {
		tmp = fma(1.0, x, (((t - a) / y) * z));
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.95e-10)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= 0.054)
		tmp = fma(1.0, x, Float64(Float64(Float64(t - a) / y) * z));
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.95e-10], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.054], N[(1.0 * x + N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.95e-10

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if -1.95e-10 < z < 0.0539999999999999994

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 35.9

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

   9. Applied rewrites 35.9%

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

   if 0.0539999999999999994 < z

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 50.9

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

   6. Applied rewrites 50.9%

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

Alternative 12: 72.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.054:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{t - a}{y} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.95e-10)
     t_1
     (if (<= z 0.054) (fma 1.0 x (* (/ (- t a) y) z)) 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 <= -1.95e-10) {
		tmp = t_1;
	} else if (z <= 0.054) {
		tmp = fma(1.0, x, (((t - a) / y) * z));
	} 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 <= -1.95e-10)
		tmp = t_1;
	elseif (z <= 0.054)
		tmp = fma(1.0, x, Float64(Float64(Float64(t - a) / y) * z));
	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, -1.95e-10], t$95$1, If[LessEqual[z, 0.054], N[(1.0 * x + N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95e-10 or 0.0539999999999999994 < z

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if -1.95e-10 < z < 0.0539999999999999994

   1. Initial program 66.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 35.9

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

   9. Applied rewrites 35.9%

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

Alternative 13: 65.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b, z, y\right)}\\ \mathbf{elif}\;z \leq 0.054:\\ \;\;\;\;-1 \cdot \frac{x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.95e-10)
     t_1
     (if (<= z -2.6e-80)
       (/ (* (- t a) z) (fma b z y))
       (if (<= z 0.054) (* -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 <= -1.95e-10) {
		tmp = t_1;
	} else if (z <= -2.6e-80) {
		tmp = ((t - a) * z) / fma(b, z, y);
	} else if (z <= 0.054) {
		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 <= -1.95e-10)
		tmp = t_1;
	elseif (z <= -2.6e-80)
		tmp = Float64(Float64(Float64(t - a) * z) / fma(b, z, y));
	elseif (z <= 0.054)
		tmp = Float64(-1.0 * Float64(x / Float64(z - 1.0)));
	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, -1.95e-10], t$95$1, If[LessEqual[z, -2.6e-80], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(b * z + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.054], N[(-1.0 * N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.95e-10 or 0.0539999999999999994 < z

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if -1.95e-10 < z < -2.6000000000000001e-80

   1. Initial program 66.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 41.6

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

   4. Applied rewrites 41.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 34.9

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

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + z \cdot b\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(+-commutative, \left(z \cdot b + y\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z \cdot b\right)\right) + y} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(*-commutative, \left(b \cdot z\right)\right) + y} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(t - a\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(b, z, y\right)\right)\right)} \]

   9. Applied rewrites 34.9%

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

   if -2.6000000000000001e-80 < z < 0.0539999999999999994

   1. Initial program 66.4%

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

   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 33.7

         \[\leadsto -1 \cdot \frac{x}{z - \color{blue}{1}} \]

   4. Applied rewrites 33.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{z - 1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.054:\\ \;\;\;\;-1 \cdot \frac{x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.1e-72) t_1 (if (<= z 0.054) (* -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 <= -1.1e-72) {
		tmp = t_1;
	} else if (z <= 0.054) {
		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 <= (-1.1d-72)) then
        tmp = t_1
    else if (z <= 0.054d0) 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 <= -1.1e-72) {
		tmp = t_1;
	} else if (z <= 0.054) {
		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 <= -1.1e-72:
		tmp = t_1
	elif z <= 0.054:
		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 <= -1.1e-72)
		tmp = t_1;
	elseif (z <= 0.054)
		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 <= -1.1e-72)
		tmp = t_1;
	elseif (z <= 0.054)
		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, -1.1e-72], t$95$1, If[LessEqual[z, 0.054], 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 < -1.10000000000000001e-72 or 0.0539999999999999994 < z

   1. Initial program 66.4%

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

   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 51.3

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

   4. Applied rewrites 51.3%

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

   if -1.10000000000000001e-72 < z < 0.0539999999999999994

   1. Initial program 66.4%

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

   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 33.7

         \[\leadsto -1 \cdot \frac{x}{z - \color{blue}{1}} \]

   4. Applied rewrites 33.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{z - 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.3% accurate, 2.5× speedup

Math

\[\frac{t - a}{b - y} \]

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 66.4%

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

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 51.3

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

4. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
5. Add Preprocessing

Alternative 16: 35.1% accurate, 3.4× speedup

Math

\[\frac{t - a}{b} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (t - a) / 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 - a) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

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 35.1

      \[\leadsto \frac{t - a}{b} \]

4. Applied rewrites 35.1%

   \[\leadsto \color{blue}{\frac{t - a}{b}} \]
5. Add Preprocessing

Alternative 17: 19.7% accurate, 5.5× speedup

Math

\[\frac{t}{b} \]

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 66.4%

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

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 35.1

      \[\leadsto \frac{t - a}{b} \]

4. Applied rewrites 35.1%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 19.7

      \[\leadsto \frac{t}{b} \]

7. Applied rewrites 19.7%

   \[\leadsto \frac{t}{\color{blue}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(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)))))