Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.7% → 88.9%

Time: 8.5s

Alternatives: 17

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: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+27} \lor \neg \left(z \leq 8.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{t - a}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{x}, t - a, y\right)}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.4e+27) (not (<= z 8.5e-9)))
   (-
    (/ (fma (/ x z) y (- t a)) (- b y))
    (* (/ y (pow (- b y) 2.0)) (/ (- t a) z)))
   (* (/ (fma (/ z x) (- t a) 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 <= -1.4e+27) || !(z <= 8.5e-9)) {
		tmp = (fma((x / z), y, (t - a)) / (b - y)) - ((y / pow((b - y), 2.0)) * ((t - a) / z));
	} else {
		tmp = (fma((z / x), (t - a), y) / fma((b - y), z, y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.4e+27) || !(z <= 8.5e-9))
		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)));
	else
		tmp = Float64(Float64(fma(Float64(z / x), Float64(t - a), y) / fma(Float64(b - y), z, y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.4e+27], N[Not[LessEqual[z, 8.5e-9]], $MachinePrecision]], 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], N[(N[(N[(N[(z / x), $MachinePrecision] * N[(t - a), $MachinePrecision] + y), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e27 or 8.5e-9 < z

   1. Initial program 40.6%

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

   3. Taylor expanded in 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 94.7%

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

   if -1.4e27 < z < 8.5e-9

   1. Initial program 82.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+27} \lor \neg \left(z \leq 8.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{t - a}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{x}, t - a, y\right)}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)}{1 - z}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-266} \lor \neg \left(t\_2 \leq 0 \lor \neg \left(t\_2 \leq 5 \cdot 10^{+289}\right)\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

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 (<= t_2 (- INFINITY))
     (/ (fma z (/ (- t a) y) x) (- 1.0 z))
     (if (or (<= t_2 -2e-266) (not (or (<= t_2 0.0) (not (<= t_2 5e+289)))))
       (/ (fma y x (* (- t a) z)) t_1)
       (/ (- 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 = ((x * y) + (z * (t - a))) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(z, ((t - a) / y), x) / (1.0 - z);
	} else if ((t_2 <= -2e-266) || !((t_2 <= 0.0) || !(t_2 <= 5e+289))) {
		tmp = fma(y, x, ((t - a) * z)) / t_1;
	} else {
		tmp = (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(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(z, Float64(Float64(t - a) / y), x) / Float64(1.0 - z));
	elseif ((t_2 <= -2e-266) || !((t_2 <= 0.0) || !(t_2 <= 5e+289)))
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / t_1);
	else
		tmp = 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[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, -2e-266], N[Not[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, 5e+289]], $MachinePrecision]]], $MachinePrecision]], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $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 19.6%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 19.6

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

   5. Applied rewrites 19.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -2e-266 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.00000000000000031e289

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      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 99.5

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

   if -2e-266 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or 5.00000000000000031e289 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 13.2%

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

   3. Taylor expanded in 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 70.6

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

   5. Applied rewrites 70.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)}{1 - z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{-266} \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0 \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{+289}\right)\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2800000000 \lor \neg \left(z \leq 7.8 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{x}, t - a, y\right)}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2800000000.0) (not (<= z 7.8e-35)))
   (/ (- t a) (- b y))
   (* (/ (fma (/ z x) (- t a) 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 <= -2800000000.0) || !(z <= 7.8e-35)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = (fma((z / x), (t - a), y) / fma((b - y), z, y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2800000000.0) || !(z <= 7.8e-35))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(Float64(fma(Float64(z / x), Float64(t - a), y) / fma(Float64(b - y), z, y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2800000000.0], N[Not[LessEqual[z, 7.8e-35]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z / x), $MachinePrecision] * N[(t - a), $MachinePrecision] + y), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e9 or 7.79999999999999961e-35 < z

   1. Initial program 41.6%

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

   3. Taylor expanded in 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 82.0

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

   5. Applied rewrites 82.0%

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

   if -2.8e9 < z < 7.79999999999999961e-35

   1. Initial program 83.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 90.1

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

   5. Applied rewrites 90.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2800000000 \lor \neg \left(z \leq 7.8 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{x}, t - a, y\right)}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -33000000000 \lor \neg \left(z \leq 1.4 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -33000000000.0) (not (<= z 1.4e-70)))
   (/ (- t a) (- b y))
   (/ (fma z (/ (- t a) y) x) (- 1.0 z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -33000000000.0) || !(z <= 1.4e-70)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(z, ((t - a) / y), x) / (1.0 - z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -33000000000.0) || !(z <= 1.4e-70))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(z, Float64(Float64(t - a) / y), x) / Float64(1.0 - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -33000000000.0], N[Not[LessEqual[z, 1.4e-70]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e10 or 1.4e-70 < z

   1. Initial program 45.2%

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

   3. Taylor expanded in 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 79.3

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

   5. Applied rewrites 79.3%

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

   if -3.3e10 < z < 1.4e-70

   1. Initial program 81.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -33000000000 \lor \neg \left(z \leq 1.4 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00029 \lor \neg \left(z \leq 1.4 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -0.00029) (not (<= z 1.4e-70)))
   (/ (- t a) (- b y))
   (/ (fma z (/ (- t a) y) x) 1.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -0.00029) || !(z <= 1.4e-70)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(z, ((t - a) / y), x) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -0.00029) || !(z <= 1.4e-70))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(z, Float64(Float64(t - a) / y), x) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -0.00029], N[Not[LessEqual[z, 1.4e-70]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e-4 or 1.4e-70 < z

   1. Initial program 45.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 78.5

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

   5. Applied rewrites 78.5%

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

   if -2.9e-4 < z < 1.4e-70

   1. Initial program 82.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 78.9%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00029 \lor \neg \left(z \leq 1.4 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5500000000 \lor \neg \left(z \leq 1.4 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{z}{y}, x\right)}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -5500000000.0) (not (<= z 1.4e-70)))
   (/ (- t a) (- b y))
   (/ (fma t (/ z y) x) (- 1.0 z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -5500000000.0) || !(z <= 1.4e-70)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(t, (z / y), x) / (1.0 - z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -5500000000.0) || !(z <= 1.4e-70))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(t, Float64(z / y), x) / Float64(1.0 - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -5500000000.0], N[Not[LessEqual[z, 1.4e-70]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(z / y), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e9 or 1.4e-70 < z

   1. Initial program 45.2%

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

   3. Taylor expanded in 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 79.3

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

   5. Applied rewrites 79.3%

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

   if -5.5e9 < z < 1.4e-70

   1. Initial program 81.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x + \frac{t \cdot z}{y}}{1 - z} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5500000000 \lor \neg \left(z \leq 1.4 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{z}{y}, x\right)}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -130:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-159}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -130.0)
     t_1
     (if (<= z -3.9e-159)
       (/ (* (- t a) z) (+ y (* z (- b y))))
       (if (<= z 1.3e-37) (* (/ y (fma (- b y) z y)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -130.0) {
		tmp = t_1;
	} else if (z <= -3.9e-159) {
		tmp = ((t - a) * z) / (y + (z * (b - y)));
	} else if (z <= 1.3e-37) {
		tmp = (y / fma((b - y), z, y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -130.0)
		tmp = t_1;
	elseif (z <= -3.9e-159)
		tmp = Float64(Float64(Float64(t - a) * z) / Float64(y + Float64(z * Float64(b - y))));
	elseif (z <= 1.3e-37)
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -130.0], t$95$1, If[LessEqual[z, -3.9e-159], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e-37], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -130 or 1.2999999999999999e-37 < z

   1. Initial program 43.0%

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

   3. Taylor expanded in 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 80.5

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

   5. Applied rewrites 80.5%

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

   if -130 < z < -3.89999999999999977e-159

   1. Initial program 84.7%

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

   3. Taylor expanded in 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 64.0

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

   5. Applied rewrites 64.0%

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

   if -3.89999999999999977e-159 < z < 1.2999999999999999e-37

   1. Initial program 82.8%

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

   3. Taylor expanded in 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 66.2

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

   5. Applied rewrites 66.2%

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

Alternative 8: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -130:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-159}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -130.0)
     t_2
     (if (<= z -3.9e-159)
       (* (- t a) (/ z t_1))
       (if (<= z 1.3e-37) (* (/ y t_1) x) t_2)))))

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 = (t - a) / (b - y);
	double tmp;
	if (z <= -130.0) {
		tmp = t_2;
	} else if (z <= -3.9e-159) {
		tmp = (t - a) * (z / t_1);
	} else if (z <= 1.3e-37) {
		tmp = (y / t_1) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -130.0)
		tmp = t_2;
	elseif (z <= -3.9e-159)
		tmp = Float64(Float64(t - a) * Float64(z / t_1));
	elseif (z <= 1.3e-37)
		tmp = Float64(Float64(y / t_1) * x);
	else
		tmp = t_2;
	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[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -130.0], t$95$2, If[LessEqual[z, -3.9e-159], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e-37], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -130 or 1.2999999999999999e-37 < z

   1. Initial program 43.0%

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

   3. Taylor expanded in 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 80.5

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

   5. Applied rewrites 80.5%

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

   if -130 < z < -3.89999999999999977e-159

   1. Initial program 84.7%

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

   3. Taylor expanded in 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 63.7

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

   5. Applied rewrites 63.7%

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

   if -3.89999999999999977e-159 < z < 1.2999999999999999e-37

   1. Initial program 82.8%

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

   3. Taylor expanded in 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 66.2

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

   5. Applied rewrites 66.2%

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

Alternative 9: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0019 \lor \neg \left(z \leq 1.3 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -0.0019) (not (<= z 1.3e-37)))
   (/ (- 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 <= -0.0019) || !(z <= 1.3e-37)) {
		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 <= -0.0019) || !(z <= 1.3e-37))
		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, -0.0019], N[Not[LessEqual[z, 1.3e-37]], $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 < -0.0019 or 1.2999999999999999e-37 < z

   1. Initial program 43.8%

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

   3. Taylor expanded in 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 80.3

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

   5. Applied rewrites 80.3%

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

   if -0.0019 < z < 1.2999999999999999e-37

   1. Initial program 82.9%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 61.0

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

   5. Applied rewrites 61.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0019 \lor \neg \left(z \leq 1.3 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0019 \lor \neg \left(z \leq 6 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -0.0019) (not (<= z 6e-71)))
   (/ (- t a) (- b y))
   (/ x (- 1.0 z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -0.0019) || !(z <= 6e-71)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x / (1.0 - z);
	}
	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 ((z <= (-0.0019d0)) .or. (.not. (z <= 6d-71))) then
        tmp = (t - a) / (b - y)
    else
        tmp = x / (1.0d0 - z)
    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 ((z <= -0.0019) || !(z <= 6e-71)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x / (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -0.0019) or not (z <= 6e-71):
		tmp = (t - a) / (b - y)
	else:
		tmp = x / (1.0 - z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -0.0019) || !(z <= 6e-71))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(x / Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -0.0019) || ~((z <= 6e-71)))
		tmp = (t - a) / (b - y);
	else
		tmp = x / (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -0.0019], N[Not[LessEqual[z, 6e-71]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0019 or 6.0000000000000003e-71 < z

   1. Initial program 45.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 78.5

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

   5. Applied rewrites 78.5%

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

   if -0.0019 < z < 6.0000000000000003e-71

   1. Initial program 82.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 56.0

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

   5. Applied rewrites 56.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0019 \lor \neg \left(z \leq 6 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-95} \lor \neg \left(y \leq 3.5 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.7e-95) (not (<= y 3.5e-29))) (/ 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 <= -1.7e-95) || !(y <= 3.5e-29)) {
		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 <= (-1.7d-95)) .or. (.not. (y <= 3.5d-29))) 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 <= -1.7e-95) || !(y <= 3.5e-29)) {
		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 <= -1.7e-95) or not (y <= 3.5e-29):
		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 <= -1.7e-95) || !(y <= 3.5e-29))
		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 <= -1.7e-95) || ~((y <= 3.5e-29)))
		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, -1.7e-95], N[Not[LessEqual[y, 3.5e-29]], $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 < -1.69999999999999997e-95 or 3.4999999999999997e-29 < y

   1. Initial program 50.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 46.9

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

   5. Applied rewrites 46.9%

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

   if -1.69999999999999997e-95 < y < 3.4999999999999997e-29

   1. Initial program 76.7%

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

   3. Taylor expanded in 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 56.1

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

   5. Applied rewrites 56.1%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-95} \lor \neg \left(y \leq 3.5 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-148} \lor \neg \left(y \leq 2.45 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.4e-148) (not (<= y 2.45e-29))) (/ x (- 1.0 z)) (/ (- a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.4e-148) || !(y <= 2.45e-29)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = -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 <= (-2.4d-148)) .or. (.not. (y <= 2.45d-29))) then
        tmp = x / (1.0d0 - z)
    else
        tmp = -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 <= -2.4e-148) || !(y <= 2.45e-29)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.4e-148) or not (y <= 2.45e-29):
		tmp = x / (1.0 - z)
	else:
		tmp = -a / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.4e-148) || !(y <= 2.45e-29))
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.4e-148) || ~((y <= 2.45e-29)))
		tmp = x / (1.0 - z);
	else
		tmp = -a / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.4e-148], N[Not[LessEqual[y, 2.45e-29]], $MachinePrecision]], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[((-a) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4000000000000001e-148 or 2.4499999999999999e-29 < y

   1. Initial program 52.9%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 44.9

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

   5. Applied rewrites 44.9%

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

   if -2.4000000000000001e-148 < y < 2.4499999999999999e-29

   1. Initial program 75.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 36.3%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-148} \lor \neg \left(y \leq 2.45 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1750000000:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1750000000.0)
   (/ (- t a) (- y))
   (if (<= z 6e-71) (/ 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 (z <= -1750000000.0) {
		tmp = (t - a) / -y;
	} else if (z <= 6e-71) {
		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 (z <= (-1750000000.0d0)) then
        tmp = (t - a) / -y
    else if (z <= 6d-71) 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 (z <= -1750000000.0) {
		tmp = (t - a) / -y;
	} else if (z <= 6e-71) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1750000000.0:
		tmp = (t - a) / -y
	elif z <= 6e-71:
		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 (z <= -1750000000.0)
		tmp = Float64(Float64(t - a) / Float64(-y));
	elseif (z <= 6e-71)
		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 (z <= -1750000000.0)
		tmp = (t - a) / -y;
	elseif (z <= 6e-71)
		tmp = x / (1.0 - z);
	else
		tmp = (t - a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1750000000.0], N[(N[(t - a), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[z, 6e-71], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75e9

   1. Initial program 41.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{t - a}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.9%

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

   if -1.75e9 < z < 6.0000000000000003e-71

   1. Initial program 82.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 55.4

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

   5. Applied rewrites 55.4%

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

   if 6.0000000000000003e-71 < z

   1. Initial program 48.1%

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

   3. Taylor expanded in y around 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 53.3

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

   5. Applied rewrites 53.3%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1750000000:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0015 \lor \neg \left(z \leq 2.3 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -0.0015) (not (<= z 2.3e-10))) (/ (- a) b) (fma z x x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -0.0015) || !(z <= 2.3e-10)) {
		tmp = -a / b;
	} else {
		tmp = fma(z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -0.0015) || !(z <= 2.3e-10))
		tmp = Float64(Float64(-a) / b);
	else
		tmp = fma(z, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -0.0015], N[Not[LessEqual[z, 2.3e-10]], $MachinePrecision]], N[((-a) / b), $MachinePrecision], N[(z * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0015 or 2.30000000000000007e-10 < z

   1. Initial program 41.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 25.3

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 27.1%

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

   if -0.0015 < z < 2.30000000000000007e-10

   1. Initial program 83.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 51.8%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0015 \lor \neg \left(z \leq 2.3 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, x, x\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y, z, t, a, b)
	return fma(z, x, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(z * x + x), $MachinePrecision]

Derivation

1. Initial program 60.9%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 33.7

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

5. Applied rewrites 33.7%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 25.7%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{x}, x\right) \]
9. Add Preprocessing

Alternative 16: 25.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a, b):
	return 1.0 * x

Julia

function code(x, y, z, t, a, b)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. div-add-rev N/A

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

   7. lower-/.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in z around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 25.6%

   \[\leadsto 1 \cdot x \]
9. Add Preprocessing

Alternative 17: 3.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ z \cdot x \end{array} \]

FPCore

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

C

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

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 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(z * x), $MachinePrecision]

Derivation

1. Initial program 60.9%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 33.7

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

5. Applied rewrites 33.7%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 25.7%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{x}, x\right) \]

9. Taylor expanded in z around inf

   \[\leadsto x \cdot z \]
10. Step-by-step derivation

11. Applied rewrites 3.8%

    \[\leadsto z \cdot x \]
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 2025017 
(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)))))