Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.7% → 98.4%

Time: 10.7s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := \mathsf{fma}\left(b - y, z, y\right)\\ t_5 := y \cdot \frac{x}{t\_4}\\ t_6 := {\left(b - y\right)}^{2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, t\_5\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-274}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{t\_6} \cdot \frac{t - a}{z}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{t\_4}, t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-x\right) \cdot \frac{y}{b - y} + y \cdot \frac{t - a}{t\_6}}{z}, -1, \frac{t - a}{b - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (fma y x (* (- t a) z)) t_1))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (fma (- b y) z y))
        (t_5 (* y (/ x t_4)))
        (t_6 (pow (- b y) 2.0)))
   (if (<= t_3 (- INFINITY))
     (fma (- t a) (pow (- b y) -1.0) t_5)
     (if (<= t_3 -1e-274)
       t_2
       (if (<= t_3 0.0)
         (- (/ (fma (/ x z) y (- t a)) (- b y)) (* (/ y t_6) (/ (- t a) z)))
         (if (<= t_3 2e+268)
           t_2
           (if (<= t_3 INFINITY)
             (fma (- t a) (/ z t_4) t_5)
             (fma
              (/ (+ (* (- x) (/ y (- b y))) (* y (/ (- t a) t_6))) z)
              -1.0
              (/ (- t a) (- b y))))))))))

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(y, x, ((t - a) * z)) / t_1;
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = fma((b - y), z, y);
	double t_5 = y * (x / t_4);
	double t_6 = pow((b - y), 2.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma((t - a), pow((b - y), -1.0), t_5);
	} else if (t_3 <= -1e-274) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = (fma((x / z), y, (t - a)) / (b - y)) - ((y / t_6) * ((t - a) / z));
	} else if (t_3 <= 2e+268) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((t - a), (z / t_4), t_5);
	} else {
		tmp = fma((((-x * (y / (b - y))) + (y * ((t - a) / t_6))) / z), -1.0, ((t - a) / (b - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = fma(Float64(b - y), z, y)
	t_5 = Float64(y * Float64(x / t_4))
	t_6 = Float64(b - y) ^ 2.0
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(Float64(t - a), (Float64(b - y) ^ -1.0), t_5);
	elseif (t_3 <= -1e-274)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(fma(Float64(x / z), y, Float64(t - a)) / Float64(b - y)) - Float64(Float64(y / t_6) * Float64(Float64(t - a) / z)));
	elseif (t_3 <= 2e+268)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = fma(Float64(t - a), Float64(z / t_4), t_5);
	else
		tmp = fma(Float64(Float64(Float64(Float64(-x) * Float64(y / Float64(b - y))) + Float64(y * Float64(Float64(t - a) / t_6))) / z), -1.0, Float64(Float64(t - a) / Float64(b - y)));
	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[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $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[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$5 = N[(y * N[(x / t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(t - a), $MachinePrecision] * N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$3, -1e-274], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(N[(N[(x / z), $MachinePrecision] * y + N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(y / t$95$6), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+268], t$95$2, If[LessEqual[t$95$3, Infinity], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$4), $MachinePrecision] + t$95$5), $MachinePrecision], N[(N[(N[(N[((-x) * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[(t - a), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -1.0 + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $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)))) < -inf.0

   1. Initial program 39.5%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999966e-275 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e268

   1. Initial program 99.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   if -9.99999999999999966e-275 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 15.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 100.0%

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

   if 1.9999999999999999e268 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 32.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 21.7

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

   5. Applied rewrites 21.7%

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

   6. Taylor expanded in z around -inf

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. div-sub N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(-x\right) \cdot \frac{y}{b - y} - \left(-y\right) \cdot \frac{t - a}{{\left(b - y\right)}^{2}}}{z}, -1, \frac{t - a}{b - y}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-274}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{t - a}{z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-x\right) \cdot \frac{y}{b - y} + y \cdot \frac{t - a}{{\left(b - y\right)}^{2}}}{z}, -1, \frac{t - a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := {\left(b - y\right)}^{-1}\\ t_5 := \mathsf{fma}\left(b - y, z, y\right)\\ t_6 := y \cdot \frac{x}{t\_5}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, t\_4, t\_6\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-274}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{t - a}{z}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{t\_5}, t\_6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - a, t\_4, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (fma y x (* (- t a) z)) t_1))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (pow (- b y) -1.0))
        (t_5 (fma (- b y) z y))
        (t_6 (* y (/ x t_5))))
   (if (<= t_3 (- INFINITY))
     (fma (- t a) t_4 t_6)
     (if (<= t_3 -1e-274)
       t_2
       (if (<= t_3 0.0)
         (-
          (/ (fma (/ x z) y (- t a)) (- b y))
          (* (/ y (pow (- b y) 2.0)) (/ (- t a) z)))
         (if (<= t_3 2e+268)
           t_2
           (if (<= t_3 INFINITY)
             (fma (- t a) (/ z t_5) t_6)
             (fma (- t a) t_4 (* (/ x z) (/ y (- b y)))))))))))

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(y, x, ((t - a) * z)) / t_1;
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = pow((b - y), -1.0);
	double t_5 = fma((b - y), z, y);
	double t_6 = y * (x / t_5);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma((t - a), t_4, t_6);
	} else if (t_3 <= -1e-274) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = (fma((x / z), y, (t - a)) / (b - y)) - ((y / pow((b - y), 2.0)) * ((t - a) / z));
	} else if (t_3 <= 2e+268) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((t - a), (z / t_5), t_6);
	} else {
		tmp = fma((t - a), t_4, ((x / z) * (y / (b - y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = Float64(b - y) ^ -1.0
	t_5 = fma(Float64(b - y), z, y)
	t_6 = Float64(y * Float64(x / t_5))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(Float64(t - a), t_4, t_6);
	elseif (t_3 <= -1e-274)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(fma(Float64(x / z), y, Float64(t - a)) / Float64(b - y)) - Float64(Float64(y / (Float64(b - y) ^ 2.0)) * Float64(Float64(t - a) / z)));
	elseif (t_3 <= 2e+268)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = fma(Float64(t - a), Float64(z / t_5), t_6);
	else
		tmp = fma(Float64(t - a), t_4, Float64(Float64(x / z) * Float64(y / Float64(b - y))));
	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[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $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[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$6 = N[(y * N[(x / t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(t - a), $MachinePrecision] * t$95$4 + t$95$6), $MachinePrecision], If[LessEqual[t$95$3, -1e-274], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(N[(N[(x / z), $MachinePrecision] * y + N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+268], t$95$2, If[LessEqual[t$95$3, Infinity], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$5), $MachinePrecision] + t$95$6), $MachinePrecision], N[(N[(t - a), $MachinePrecision] * t$95$4 + N[(N[(x / z), $MachinePrecision] * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)))) < -inf.0

   1. Initial program 39.5%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999966e-275 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e268

   1. Initial program 99.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   if -9.99999999999999966e-275 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 15.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 100.0%

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

   if 1.9999999999999999e268 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 32.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

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

   1. Initial program 0.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 8.9

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

   4. Applied rewrites 8.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 74.0

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

   7. Applied rewrites 74.0%

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

   8. Taylor expanded in z around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.5

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

   10. Applied rewrites 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-274}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{t - a}{z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := {\left(b - y\right)}^{-1}\\ t_5 := \mathsf{fma}\left(t - a, t\_4, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ t_6 := \mathsf{fma}\left(b - y, z, y\right)\\ t_7 := y \cdot \frac{x}{t\_6}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, t\_4, t\_7\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-274}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{t\_6}, t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (fma y x (* (- t a) z)) t_1))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (pow (- b y) -1.0))
        (t_5 (fma (- t a) t_4 (* (/ x z) (/ y (- b y)))))
        (t_6 (fma (- b y) z y))
        (t_7 (* y (/ x t_6))))
   (if (<= t_3 (- INFINITY))
     (fma (- t a) t_4 t_7)
     (if (<= t_3 -1e-274)
       t_2
       (if (<= t_3 0.0)
         t_5
         (if (<= t_3 2e+268)
           t_2
           (if (<= t_3 INFINITY) (fma (- t a) (/ z t_6) t_7) 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(y, x, ((t - a) * z)) / t_1;
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = pow((b - y), -1.0);
	double t_5 = fma((t - a), t_4, ((x / z) * (y / (b - y))));
	double t_6 = fma((b - y), z, y);
	double t_7 = y * (x / t_6);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma((t - a), t_4, t_7);
	} else if (t_3 <= -1e-274) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_5;
	} else if (t_3 <= 2e+268) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((t - a), (z / t_6), t_7);
	} 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 = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = Float64(b - y) ^ -1.0
	t_5 = fma(Float64(t - a), t_4, Float64(Float64(x / z) * Float64(y / Float64(b - y))))
	t_6 = fma(Float64(b - y), z, y)
	t_7 = Float64(y * Float64(x / t_6))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(Float64(t - a), t_4, t_7);
	elseif (t_3 <= -1e-274)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = t_5;
	elseif (t_3 <= 2e+268)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = fma(Float64(t - a), Float64(z / t_6), t_7);
	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[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $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[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t - a), $MachinePrecision] * t$95$4 + N[(N[(x / z), $MachinePrecision] * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$7 = N[(y * N[(x / t$95$6), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(t - a), $MachinePrecision] * t$95$4 + t$95$7), $MachinePrecision], If[LessEqual[t$95$3, -1e-274], t$95$2, If[LessEqual[t$95$3, 0.0], t$95$5, If[LessEqual[t$95$3, 2e+268], t$95$2, If[LessEqual[t$95$3, Infinity], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$6), $MachinePrecision] + t$95$7), $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

   1. Initial program 39.5%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999966e-275 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e268

   1. Initial program 99.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   if -9.99999999999999966e-275 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 3.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 10.4

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

   4. Applied rewrites 10.4%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in z around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if 1.9999999999999999e268 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 32.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-274}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ t_3 := {\left(b - y\right)}^{-1}\\ t_4 := \mathsf{fma}\left(t - a, t\_3, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ t_5 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_6 := \mathsf{fma}\left(t - a, t\_3, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-274}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (fma y x (* (- t a) z)) t_1))
        (t_3 (pow (- b y) -1.0))
        (t_4 (fma (- t a) t_3 (* (/ x z) (/ y (- b y)))))
        (t_5 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_6 (fma (- t a) t_3 (* y (/ x (fma (- b y) z y))))))
   (if (<= t_5 (- INFINITY))
     t_6
     (if (<= t_5 -1e-274)
       t_2
       (if (<= t_5 0.0)
         t_4
         (if (<= t_5 2e+268) t_2 (if (<= t_5 INFINITY) t_6 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 = fma(y, x, ((t - a) * z)) / t_1;
	double t_3 = pow((b - y), -1.0);
	double t_4 = fma((t - a), t_3, ((x / z) * (y / (b - y))));
	double t_5 = ((x * y) + (z * (t - a))) / t_1;
	double t_6 = fma((t - a), t_3, (y * (x / fma((b - y), z, y))));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_6;
	} else if (t_5 <= -1e-274) {
		tmp = t_2;
	} else if (t_5 <= 0.0) {
		tmp = t_4;
	} else if (t_5 <= 2e+268) {
		tmp = t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_6;
	} 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(fma(y, x, Float64(Float64(t - a) * z)) / t_1)
	t_3 = Float64(b - y) ^ -1.0
	t_4 = fma(Float64(t - a), t_3, Float64(Float64(x / z) * Float64(y / Float64(b - y))))
	t_5 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_6 = fma(Float64(t - a), t_3, Float64(y * Float64(x / fma(Float64(b - y), z, y))))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_6;
	elseif (t_5 <= -1e-274)
		tmp = t_2;
	elseif (t_5 <= 0.0)
		tmp = t_4;
	elseif (t_5 <= 2e+268)
		tmp = t_2;
	elseif (t_5 <= Inf)
		tmp = t_6;
	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[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t - a), $MachinePrecision] * t$95$3 + N[(N[(x / z), $MachinePrecision] * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t - a), $MachinePrecision] * t$95$3 + N[(y * N[(x / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], t$95$6, If[LessEqual[t$95$5, -1e-274], t$95$2, If[LessEqual[t$95$5, 0.0], t$95$4, If[LessEqual[t$95$5, 2e+268], t$95$2, If[LessEqual[t$95$5, Infinity], t$95$6, t$95$4]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.9999999999999999e268 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 35.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.5

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

   7. Applied rewrites 96.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999966e-275 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e268

   1. Initial program 99.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   if -9.99999999999999966e-275 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 3.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 10.4

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

   4. Applied rewrites 10.4%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in z around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.6

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

   10. Applied rewrites 99.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-274}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty \lor \neg \left(t\_2 \leq 2 \cdot 10^{+268}\right):\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ \end{array} \end{array} \]

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)))
   (if (or (<= t_2 (- INFINITY)) (not (<= t_2 2e+268)))
     (fma (- t a) (pow (- b y) -1.0) (* y (/ x (fma (- b y) z y))))
     (/ (fma y x (* (- t a) z)) t_1))))

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 tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 2e+268)) {
		tmp = fma((t - a), pow((b - y), -1.0), (y * (x / fma((b - y), z, y))));
	} else {
		tmp = fma(y, x, ((t - a) * z)) / t_1;
	}
	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)
	tmp = 0.0
	if ((t_2 <= Float64(-Inf)) || !(t_2 <= 2e+268))
		tmp = fma(Float64(t - a), (Float64(b - y) ^ -1.0), Float64(y * Float64(x / fma(Float64(b - y), z, y))));
	else
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1);
	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]}, If[Or[LessEqual[t$95$2, (-Infinity)], N[Not[LessEqual[t$95$2, 2e+268]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] * N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision] + N[(y * N[(x / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.9999999999999999e268 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 22.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 88.5

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

   7. Applied rewrites 88.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e268

   1. Initial program 94.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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 94.4

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

      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 94.4

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

   4. Applied rewrites 94.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+268}\right):\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_3 := {\left(b - y\right)}^{-1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, t\_3, y \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - a, t\_3, \frac{-x}{z}\right)\\ \end{array} \end{array} \]

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 (pow (- b y) -1.0)))
   (if (<= t_2 (- INFINITY))
     (fma (- t a) t_3 (* y (/ x y)))
     (if (<= t_2 2e+268)
       (/ (fma y x (* (- t a) z)) t_1)
       (fma (- t a) t_3 (/ (- x) z))))))

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 = pow((b - y), -1.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((t - a), t_3, (y * (x / y)));
	} else if (t_2 <= 2e+268) {
		tmp = fma(y, x, ((t - a) * z)) / t_1;
	} else {
		tmp = fma((t - a), t_3, (-x / z));
	}
	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 = Float64(b - y) ^ -1.0
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(t - a), t_3, Float64(y * Float64(x / y)));
	elseif (t_2 <= 2e+268)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1);
	else
		tmp = fma(Float64(t - a), t_3, Float64(Float64(-x) / z));
	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[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t - a), $MachinePrecision] * t$95$3 + N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+268], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(t - a), $MachinePrecision] * t$95$3 + N[((-x) / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.5%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 86.3

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

   10. Applied rewrites 86.3%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e268

   1. Initial program 94.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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 94.4

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

      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 94.4

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

   4. Applied rewrites 94.4%

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

   if 1.9999999999999999e268 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 15.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 52.2

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 83.8

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

   7. Applied rewrites 83.8%

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

   8. Taylor expanded in z around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 79.3

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

   10. Applied rewrites 79.3%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 71.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, \frac{-x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+44} \lor \neg \left(z \leq 5.2 \cdot 10^{+32}\right):\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, \frac{-x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -5e+44) (not (<= z 5.2e+32)))
   (fma (- t a) (pow (- b y) -1.0) (/ (- x) z))
   (/ (fma y x (* (- t a) z)) (+ y (* z (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -5e+44) || !(z <= 5.2e+32)) {
		tmp = fma((t - a), pow((b - y), -1.0), (-x / z));
	} else {
		tmp = fma(y, x, ((t - a) * z)) / (y + (z * (b - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -5e+44) || !(z <= 5.2e+32))
		tmp = fma(Float64(t - a), (Float64(b - y) ^ -1.0), Float64(Float64(-x) / z));
	else
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(y + Float64(z * Float64(b - y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -5e+44], N[Not[LessEqual[z, 5.2e+32]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] * N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9999999999999996e44 or 5.2000000000000004e32 < z

   1. Initial program 42.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.6

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in z around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.7

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 88.4%

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

   if -4.9999999999999996e44 < z < 5.2000000000000004e32

   1. Initial program 85.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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 85.6

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

      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 85.6

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

   4. Applied rewrites 85.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+44} \lor \neg \left(z \leq 5.2 \cdot 10^{+32}\right):\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, \frac{-x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{y}{t\_1} \cdot x\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+33}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-121}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-242}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-21}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (* (/ y t_1) x))
        (t_3 (/ (- t a) (- b y)))
        (t_4 (/ (* z (- t a)) t_1)))
   (if (<= z -3.4e+33)
     t_3
     (if (<= z -1.55e+24)
       t_2
       (if (<= z -3.1e-121)
         t_4
         (if (<= z 4.2e-242) t_2 (if (<= z 6e-21) t_4 t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (y / t_1) * x;
	double t_3 = (t - a) / (b - y);
	double t_4 = (z * (t - a)) / t_1;
	double tmp;
	if (z <= -3.4e+33) {
		tmp = t_3;
	} else if (z <= -1.55e+24) {
		tmp = t_2;
	} else if (z <= -3.1e-121) {
		tmp = t_4;
	} else if (z <= 4.2e-242) {
		tmp = t_2;
	} else if (z <= 6e-21) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(y / t_1) * x)
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	t_4 = Float64(Float64(z * Float64(t - a)) / t_1)
	tmp = 0.0
	if (z <= -3.4e+33)
		tmp = t_3;
	elseif (z <= -1.55e+24)
		tmp = t_2;
	elseif (z <= -3.1e-121)
		tmp = t_4;
	elseif (z <= 4.2e-242)
		tmp = t_2;
	elseif (z <= 6e-21)
		tmp = t_4;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[z, -3.4e+33], t$95$3, If[LessEqual[z, -1.55e+24], t$95$2, If[LessEqual[z, -3.1e-121], t$95$4, If[LessEqual[z, 4.2e-242], t$95$2, If[LessEqual[z, 6e-21], t$95$4, t$95$3]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3999999999999999e33 or 5.99999999999999982e-21 < z

   1. Initial program 48.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.3999999999999999e33 < z < -1.55000000000000005e24 or -3.0999999999999998e-121 < z < 4.20000000000000037e-242

   1. Initial program 80.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -1.55000000000000005e24 < z < -3.0999999999999998e-121 or 4.20000000000000037e-242 < z < 5.99999999999999982e-21

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 64.2

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

   5. Applied rewrites 64.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

   7. Applied rewrites 64.2%

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

Alternative 9: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{y}{t\_1} \cdot x\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \left(t - a\right) \cdot \frac{z}{t\_1}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+33}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-121}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-234}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-21}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (* (/ y t_1) x))
        (t_3 (/ (- t a) (- b y)))
        (t_4 (* (- t a) (/ z t_1))))
   (if (<= z -3.4e+33)
     t_3
     (if (<= z -1.55e+24)
       t_2
       (if (<= z -3.1e-121)
         t_4
         (if (<= z 4.7e-234) t_2 (if (<= z 6e-21) t_4 t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (y / t_1) * x;
	double t_3 = (t - a) / (b - y);
	double t_4 = (t - a) * (z / t_1);
	double tmp;
	if (z <= -3.4e+33) {
		tmp = t_3;
	} else if (z <= -1.55e+24) {
		tmp = t_2;
	} else if (z <= -3.1e-121) {
		tmp = t_4;
	} else if (z <= 4.7e-234) {
		tmp = t_2;
	} else if (z <= 6e-21) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(y / t_1) * x)
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	t_4 = Float64(Float64(t - a) * Float64(z / t_1))
	tmp = 0.0
	if (z <= -3.4e+33)
		tmp = t_3;
	elseif (z <= -1.55e+24)
		tmp = t_2;
	elseif (z <= -3.1e-121)
		tmp = t_4;
	elseif (z <= 4.7e-234)
		tmp = t_2;
	elseif (z <= 6e-21)
		tmp = t_4;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.4e+33], t$95$3, If[LessEqual[z, -1.55e+24], t$95$2, If[LessEqual[z, -3.1e-121], t$95$4, If[LessEqual[z, 4.7e-234], t$95$2, If[LessEqual[z, 6e-21], t$95$4, t$95$3]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3999999999999999e33 or 5.99999999999999982e-21 < z

   1. Initial program 48.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.3999999999999999e33 < z < -1.55000000000000005e24 or -3.0999999999999998e-121 < z < 4.7000000000000001e-234

   1. Initial program 80.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -1.55000000000000005e24 < z < -3.0999999999999998e-121 or 4.7000000000000001e-234 < z < 5.99999999999999982e-21

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 62.9

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

   5. Applied rewrites 62.9%

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

Alternative 10: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+71} \lor \neg \left(z \leq 2.4 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.5e+71) (not (<= z 2.4e+99)))
   (/ (- t a) (- b y))
   (/ (fma y x (* (- t a) z)) (+ y (* z (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.5e+71) || !(z <= 2.4e+99)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(y, x, ((t - a) * z)) / (y + (z * (b - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.5e+71) || !(z <= 2.4e+99))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(y + Float64(z * Float64(b - y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.5e+71], N[Not[LessEqual[z, 2.4e+99]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000006e71 or 2.4000000000000001e99 < z

   1. Initial program 34.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -1.50000000000000006e71 < z < 2.4000000000000001e99

   1. Initial program 84.9%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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 84.9

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

      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 84.9

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

   4. Applied rewrites 84.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+71} \lor \neg \left(z \leq 2.4 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-26} \lor \neg \left(z \leq 6 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.4e-26) (not (<= z 6e-21)))
   (/ (- t a) (- b y))
   (/ (fma x y (* z t)) (fma (- b y) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.4e-26) || !(z <= 6e-21)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, y, (z * t)) / fma((b - y), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.4e-26) || !(z <= 6e-21))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(x, y, Float64(z * t)) / fma(Float64(b - y), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.4e-26], N[Not[LessEqual[z, 6e-21]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4000000000000001e-26 or 5.99999999999999982e-21 < z

   1. Initial program 50.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -2.4000000000000001e-26 < z < 5.99999999999999982e-21

   1. Initial program 87.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 45.5

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

   5. Applied rewrites 45.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-26} \lor \neg \left(z \leq 6 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-26} \lor \neg \left(z \leq 6 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.4e-26) (not (<= z 6e-21)))
   (/ (- t a) (- b y))
   (/ (fma t z (* y x)) (fma (- b y) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.4e-26) || !(z <= 6e-21)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(t, z, (y * x)) / fma((b - y), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.4e-26) || !(z <= 6e-21))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(t, z, Float64(y * x)) / fma(Float64(b - y), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.4e-26], N[Not[LessEqual[z, 6e-21]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4000000000000001e-26 or 5.99999999999999982e-21 < z

   1. Initial program 50.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -2.4000000000000001e-26 < z < 5.99999999999999982e-21

   1. Initial program 87.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 67.6

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

   5. Applied rewrites 67.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-26} \lor \neg \left(z \leq 6 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ t_2 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{+78}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-204}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;y \leq 98000000000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))) (t_2 (/ x (- 1.0 z))))
   (if (<= y -5.1e+78)
     t_2
     (if (<= y 2e-307)
       t_1
       (if (<= y 1.9e-204)
         (/ (- a) b)
         (if (<= y 98000000000000.0) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double t_2 = x / (1.0 - z);
	double tmp;
	if (y <= -5.1e+78) {
		tmp = t_2;
	} else if (y <= 2e-307) {
		tmp = t_1;
	} else if (y <= 1.9e-204) {
		tmp = -a / b;
	} else if (y <= 98000000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = t / (b - y)
    t_2 = x / (1.0d0 - z)
    if (y <= (-5.1d+78)) then
        tmp = t_2
    else if (y <= 2d-307) then
        tmp = t_1
    else if (y <= 1.9d-204) then
        tmp = -a / b
    else if (y <= 98000000000000.0d0) then
        tmp = t_1
    else
        tmp = t_2
    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 / (b - y);
	double t_2 = x / (1.0 - z);
	double tmp;
	if (y <= -5.1e+78) {
		tmp = t_2;
	} else if (y <= 2e-307) {
		tmp = t_1;
	} else if (y <= 1.9e-204) {
		tmp = -a / b;
	} else if (y <= 98000000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	t_2 = x / (1.0 - z)
	tmp = 0
	if y <= -5.1e+78:
		tmp = t_2
	elif y <= 2e-307:
		tmp = t_1
	elif y <= 1.9e-204:
		tmp = -a / b
	elif y <= 98000000000000.0:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	t_2 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -5.1e+78)
		tmp = t_2;
	elseif (y <= 2e-307)
		tmp = t_1;
	elseif (y <= 1.9e-204)
		tmp = Float64(Float64(-a) / b);
	elseif (y <= 98000000000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	t_2 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -5.1e+78)
		tmp = t_2;
	elseif (y <= 2e-307)
		tmp = t_1;
	elseif (y <= 1.9e-204)
		tmp = -a / b;
	elseif (y <= 98000000000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.1e+78], t$95$2, If[LessEqual[y, 2e-307], t$95$1, If[LessEqual[y, 1.9e-204], N[((-a) / b), $MachinePrecision], If[LessEqual[y, 98000000000000.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.10000000000000031e78 or 9.8e13 < y

   1. Initial program 49.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -5.10000000000000031e78 < y < 1.99999999999999982e-307 or 1.89999999999999991e-204 < y < 9.8e13

   1. Initial program 78.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.7%

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

   if 1.99999999999999982e-307 < y < 1.89999999999999991e-204

   1. Initial program 87.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 87.3

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

   4. Applied rewrites 87.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 72.7

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in a around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{a}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.8%

       \[\leadsto \frac{-a}{\color{blue}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-204}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;y \leq 98000000000000:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{-54} \lor \neg \left(z \leq 1.3 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -5.3e-54) (not (<= z 1.3e-44)))
   (/ (- t a) (- b y))
   (* (/ y (fma (- b y) z y)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -5.3e-54) || !(z <= 1.3e-44)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = (y / fma((b - y), z, y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -5.3e-54) || !(z <= 1.3e-44))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -5.3e-54], N[Not[LessEqual[z, 1.3e-44]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.30000000000000057e-54 or 1.2999999999999999e-44 < z

   1. Initial program 55.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if -5.30000000000000057e-54 < z < 1.2999999999999999e-44

   1. Initial program 85.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 57.2

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

   5. Applied rewrites 57.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{-54} \lor \neg \left(z \leq 1.3 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+79} \lor \neg \left(y \leq 3.2 \cdot 10^{+127}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.2e+79) (not (<= y 3.2e+127)))
   (/ x (- 1.0 z))
   (/ (- t a) (- b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.2e+79) || !(y <= 3.2e+127)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / (b - y);
	}
	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) :: tmp
    if ((y <= (-1.2d+79)) .or. (.not. (y <= 3.2d+127))) then
        tmp = x / (1.0d0 - z)
    else
        tmp = (t - a) / (b - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.2e+79) || !(y <= 3.2e+127)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.2e+79) or not (y <= 3.2e+127):
		tmp = x / (1.0 - z)
	else:
		tmp = (t - a) / (b - y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.2e+79) || !(y <= 3.2e+127))
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.2e+79) || ~((y <= 3.2e+127)))
		tmp = x / (1.0 - z);
	else
		tmp = (t - a) / (b - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.2e+79], N[Not[LessEqual[y, 3.2e+127]], $MachinePrecision]], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999993e79 or 3.19999999999999976e127 < y

   1. Initial program 46.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -1.19999999999999993e79 < y < 3.19999999999999976e127

   1. Initial program 78.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 63.6

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

   5. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+79} \lor \neg \left(y \leq 3.2 \cdot 10^{+127}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+78} \lor \neg \left(y \leq 52000000000000\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.8e+78) (not (<= y 52000000000000.0)))
   (/ x (- 1.0 z))
   (/ (- t a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.8e+78) || !(y <= 52000000000000.0)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	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) :: tmp
    if ((y <= (-4.8d+78)) .or. (.not. (y <= 52000000000000.0d0))) then
        tmp = x / (1.0d0 - z)
    else
        tmp = (t - a) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.8e+78) || !(y <= 52000000000000.0)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.8e+78) or not (y <= 52000000000000.0):
		tmp = x / (1.0 - z)
	else:
		tmp = (t - a) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.8e+78) || !(y <= 52000000000000.0))
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(Float64(t - a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.8e+78) || ~((y <= 52000000000000.0)))
		tmp = x / (1.0 - z);
	else
		tmp = (t - a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.8e+78], N[Not[LessEqual[y, 52000000000000.0]], $MachinePrecision]], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7999999999999997e78 or 5.2e13 < y

   1. Initial program 49.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -4.7999999999999997e78 < y < 5.2e13

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 51.2

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

   5. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+78} \lor \neg \left(y \leq 52000000000000\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-54} \lor \neg \left(z \leq 1.4 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -5.8e-54) (not (<= z 1.4e-44))) (/ t (- b y)) (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -5.8e-54) || !(z <= 1.4e-44)) {
		tmp = t / (b - y);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -5.8e-54) || !(z <= 1.4e-44))
		tmp = Float64(t / Float64(b - y));
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -5.8e-54], N[Not[LessEqual[z, 1.4e-44]], $MachinePrecision]], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.80000000000000029e-54 or 1.4e-44 < z

   1. Initial program 55.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.9%

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

   if -5.80000000000000029e-54 < z < 1.4e-44

   1. Initial program 85.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.1%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-54} \lor \neg \left(z \leq 1.4 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-47} \lor \neg \left(z \leq 1.3 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.25e-47) (not (<= z 1.3e-44))) (/ t b) (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.25e-47) || !(z <= 1.3e-44)) {
		tmp = t / b;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.25e-47) || !(z <= 1.3e-44))
		tmp = Float64(t / b);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.25e-47], N[Not[LessEqual[z, 1.3e-44]], $MachinePrecision]], N[(t / b), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000003e-47 or 1.2999999999999999e-44 < z

   1. Initial program 54.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 26.2%

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

   if -1.25000000000000003e-47 < z < 1.2999999999999999e-44

   1. Initial program 85.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 47.7

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.7%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-47} \lor \neg \left(z \leq 1.3 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.6e-33) (/ (- a) b) (if (<= z 1.3e-44) (fma x z x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.6e-33) {
		tmp = -a / b;
	} else if (z <= 1.3e-44) {
		tmp = fma(x, z, x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.6e-33)
		tmp = Float64(Float64(-a) / b);
	elseif (z <= 1.3e-44)
		tmp = fma(x, z, x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.6e-33], N[((-a) / b), $MachinePrecision], If[LessEqual[z, 1.3e-44], N[(x * z + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.59999999999999971e-33

   1. Initial program 48.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 70.8

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 50.8

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

   7. Applied rewrites 50.8%

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

   8. Taylor expanded in a around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{a}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.8%

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

   if -4.59999999999999971e-33 < z < 1.2999999999999999e-44

   1. Initial program 86.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.7%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]

   if 1.2999999999999999e-44 < z

   1. Initial program 57.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 28.4%

      \[\leadsto \frac{t}{\color{blue}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 25.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, z, x\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y, z, t, a, b)
	return fma(x, z, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * z + x), $MachinePrecision]

Derivation

1. Initial program 67.8%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower--.f64 31.4

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

5. Applied rewrites 31.4%

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

6. Taylor expanded in z around 0

   \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation

8. Applied rewrites 23.6%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
9. Add Preprocessing

Alternative 21: 3.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ x \cdot z \end{array} \]

FPCore

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

C

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

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 * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * z), $MachinePrecision]

Derivation

1. Initial program 67.8%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower--.f64 31.4

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

5. Applied rewrites 31.4%

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

6. Taylor expanded in z around 0

   \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation

8. Applied rewrites 23.6%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]

9. Taylor expanded in z around inf

   \[\leadsto x \cdot z \]
10. Step-by-step derivation

11. Applied rewrites 3.1%

    \[\leadsto x \cdot z \]
12. Add Preprocessing

Developer Target 1: 73.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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 = (((z * t) + (y * x)) / (y + (z * (b - y)))) - (a / ((b - y) + (y / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024352 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z)))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))