Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.6% → 90.7%

Time: 12.3s

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: 66.6% 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: 90.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{\frac{y}{b - y} \cdot \left(x - t\_1\right)}{z} + t\_1\\ \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (+ (/ (* (/ y (- b y)) (- x t_1)) z) t_1)))
   (if (<= z -29000000.0)
     t_2
     (if (<= z 6.7e-19) (/ (fma y x (* (- t a) z)) (+ y (* z (- b y)))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = (((y / (b - y)) * (x - t_1)) / z) + t_1;
	double tmp;
	if (z <= -29000000.0) {
		tmp = t_2;
	} else if (z <= 6.7e-19) {
		tmp = fma(y, x, ((t - a) * z)) / (y + (z * (b - y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(Float64(y / Float64(b - y)) * Float64(x - t_1)) / z) + t_1)
	tmp = 0.0
	if (z <= -29000000.0)
		tmp = t_2;
	elseif (z <= 6.7e-19)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x - t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[z, -29000000.0], t$95$2, If[LessEqual[z, 6.7e-19], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e7 or 6.69999999999999998e-19 < z

   1. Initial program 46.5%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 94.3%

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

   if -2.9e7 < z < 6.69999999999999998e-19

   1. Initial program 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 87.2

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

   3. Applied rewrites 87.2%

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

Alternative 2: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -9.2e+45)
     t_1
     (if (<= z 1.32e+37)
       (/ (fma y x (* (- t a) z)) (+ y (* z (- b y))))
       (+ (/ (- x) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -9.2e+45) {
		tmp = t_1;
	} else if (z <= 1.32e+37) {
		tmp = fma(y, x, ((t - a) * z)) / (y + (z * (b - y)));
	} else {
		tmp = (-x / z) + 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 <= -9.2e+45)
		tmp = t_1;
	elseif (z <= 1.32e+37)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(Float64(Float64(-x) / z) + 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, -9.2e+45], t$95$1, If[LessEqual[z, 1.32e+37], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.20000000000000049e45

   1. Initial program 39.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 84.0

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

   4. Applied rewrites 84.0%

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

   if -9.20000000000000049e45 < z < 1.3199999999999999e37

   1. Initial program 86.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 86.4

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

   3. Applied rewrites 86.4%

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

   if 1.3199999999999999e37 < z

   1. Initial program 41.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \frac{-1 \cdot x}{z} + \frac{\color{blue}{t} - a}{b - y} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.1

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

   7. Applied rewrites 87.1%

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

Alternative 3: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -7 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{y \cdot x - a \cdot z}{t\_1}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{z}{t\_1} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z} + t\_2\\ \end{array} \end{array} \]

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 -7e+45)
     t_2
     (if (<= z 2.6e-146)
       (/ (- (* y x) (* a z)) t_1)
       (if (<= z 2.8e+40) (* (/ z t_1) (- t a)) (+ (/ (- x) z) 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 <= -7e+45) {
		tmp = t_2;
	} else if (z <= 2.6e-146) {
		tmp = ((y * x) - (a * z)) / t_1;
	} else if (z <= 2.8e+40) {
		tmp = (z / t_1) * (t - a);
	} else {
		tmp = (-x / z) + 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 <= -7e+45)
		tmp = t_2;
	elseif (z <= 2.6e-146)
		tmp = Float64(Float64(Float64(y * x) - Float64(a * z)) / t_1);
	elseif (z <= 2.8e+40)
		tmp = Float64(Float64(z / t_1) * Float64(t - a));
	else
		tmp = Float64(Float64(Float64(-x) / z) + 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, -7e+45], t$95$2, If[LessEqual[z, 2.6e-146], N[(N[(N[(y * x), $MachinePrecision] - N[(a * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 2.8e+40], N[(N[(z / t$95$1), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.00000000000000046e45

   1. Initial program 39.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 84.0

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

   4. Applied rewrites 84.0%

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

   if -7.00000000000000046e45 < z < 2.59999999999999987e-146

   1. Initial program 86.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. fp-cancel-sub-sign N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 65.5

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

   4. Applied rewrites 65.5%

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

   if 2.59999999999999987e-146 < z < 2.8000000000000001e40

   1. Initial program 87.2%

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

   2. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

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

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. div-add-rev N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-neg-in N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

   4. Applied rewrites 56.5%

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

   if 2.8000000000000001e40 < z

   1. Initial program 40.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \frac{-1 \cdot x}{z} + \frac{\color{blue}{t} - a}{b - y} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.3

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

   7. Applied rewrites 87.3%

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

Alternative 4: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t - a, z, y \cdot x\right)}{b \cdot z}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3e+26)
     t_1
     (if (<= z -4.1e-141)
       (/ (fma (- t a) z (* y x)) (* b z))
       (if (<= z 6.6e-142)
         (/ (+ (* x y) (* z (- t a))) y)
         (if (<= z 2.8e+40)
           (* (/ z (fma (- b y) z y)) (- t a))
           (+ (/ (- x) z) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3e+26) {
		tmp = t_1;
	} else if (z <= -4.1e-141) {
		tmp = fma((t - a), z, (y * x)) / (b * z);
	} else if (z <= 6.6e-142) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} else if (z <= 2.8e+40) {
		tmp = (z / fma((b - y), z, y)) * (t - a);
	} else {
		tmp = (-x / z) + 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 <= -3e+26)
		tmp = t_1;
	elseif (z <= -4.1e-141)
		tmp = Float64(fma(Float64(t - a), z, Float64(y * x)) / Float64(b * z));
	elseif (z <= 6.6e-142)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / y);
	elseif (z <= 2.8e+40)
		tmp = Float64(Float64(z / fma(Float64(b - y), z, y)) * Float64(t - a));
	else
		tmp = Float64(Float64(Float64(-x) / z) + 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, -3e+26], t$95$1, If[LessEqual[z, -4.1e-141], N[(N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(b * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.6e-142], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 2.8e+40], N[(N[(z / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.99999999999999997e26

   1. Initial program 41.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 82.6

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

   4. Applied rewrites 82.6%

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

   if -2.99999999999999997e26 < z < -4.10000000000000002e-141

   1. Initial program 85.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 42.3

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

   4. Applied rewrites 42.3%

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

   if -4.10000000000000002e-141 < z < 6.5999999999999994e-142

   1. Initial program 87.1%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 68.2%

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

   if 6.5999999999999994e-142 < z < 2.8000000000000001e40

   1. Initial program 87.2%

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

   2. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

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

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. div-add-rev N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-neg-in N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

   4. Applied rewrites 56.7%

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

   if 2.8000000000000001e40 < z

   1. Initial program 40.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \frac{-1 \cdot x}{z} + \frac{\color{blue}{t} - a}{b - y} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.3

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

   7. Applied rewrites 87.3%

      \[\leadsto \frac{-x}{z} + \frac{\color{blue}{t} - a}{b - y} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 70.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{z}, t - a\right)}{b}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.3e+26)
     t_1
     (if (<= z -4.1e-141)
       (/ (fma x (/ y z) (- t a)) b)
       (if (<= z 6.6e-142)
         (/ (+ (* x y) (* z (- t a))) y)
         (if (<= z 2.8e+40)
           (* (/ z (fma (- b y) z y)) (- t a))
           (+ (/ (- x) z) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.3e+26) {
		tmp = t_1;
	} else if (z <= -4.1e-141) {
		tmp = fma(x, (y / z), (t - a)) / b;
	} else if (z <= 6.6e-142) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} else if (z <= 2.8e+40) {
		tmp = (z / fma((b - y), z, y)) * (t - a);
	} else {
		tmp = (-x / z) + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.3e+26)
		tmp = t_1;
	elseif (z <= -4.1e-141)
		tmp = Float64(fma(x, Float64(y / z), Float64(t - a)) / b);
	elseif (z <= 6.6e-142)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / y);
	elseif (z <= 2.8e+40)
		tmp = Float64(Float64(z / fma(Float64(b - y), z, y)) * Float64(t - a));
	else
		tmp = Float64(Float64(Float64(-x) / z) + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+26], t$95$1, If[LessEqual[z, -4.1e-141], N[(N[(x * N[(y / z), $MachinePrecision] + N[(t - a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 6.6e-142], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 2.8e+40], N[(N[(z / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -3.29999999999999993e26

   1. Initial program 41.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 82.6

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

   4. Applied rewrites 82.6%

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

   if -3.29999999999999993e26 < z < -4.10000000000000002e-141

   1. Initial program 85.7%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{z}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x}{z} \]

      3. lower-/.f64 7.5

         \[\leadsto -\frac{x}{z} \]

   7. Applied rewrites 7.5%

      \[\leadsto -\frac{x}{z} \]

   8. Taylor expanded in b around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 41.8

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

   10. Applied rewrites 41.8%

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

   if -4.10000000000000002e-141 < z < 6.5999999999999994e-142

   1. Initial program 87.1%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 68.2%

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

   if 6.5999999999999994e-142 < z < 2.8000000000000001e40

   1. Initial program 87.2%

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

   2. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

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

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. div-add-rev N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-neg-in N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

   4. Applied rewrites 56.7%

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

   if 2.8000000000000001e40 < z

   1. Initial program 40.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \frac{-1 \cdot x}{z} + \frac{\color{blue}{t} - a}{b - y} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.3

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

   7. Applied rewrites 87.3%

      \[\leadsto \frac{-x}{z} + \frac{\color{blue}{t} - a}{b - y} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 70.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{z}, t - a\right)}{b}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y}\\ \mathbf{elif}\;z \leq 1.15:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.3e+26)
     t_1
     (if (<= z -4.1e-141)
       (/ (fma x (/ y z) (- t a)) b)
       (if (<= z 6.6e-142)
         (/ (+ (* x y) (* z (- t a))) y)
         (if (<= z 1.15)
           (* (/ z (fma b z y)) (- t a))
           (+ (/ (- x) z) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.3e+26) {
		tmp = t_1;
	} else if (z <= -4.1e-141) {
		tmp = fma(x, (y / z), (t - a)) / b;
	} else if (z <= 6.6e-142) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} else if (z <= 1.15) {
		tmp = (z / fma(b, z, y)) * (t - a);
	} else {
		tmp = (-x / z) + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.3e+26)
		tmp = t_1;
	elseif (z <= -4.1e-141)
		tmp = Float64(fma(x, Float64(y / z), Float64(t - a)) / b);
	elseif (z <= 6.6e-142)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / y);
	elseif (z <= 1.15)
		tmp = Float64(Float64(z / fma(b, z, y)) * Float64(t - a));
	else
		tmp = Float64(Float64(Float64(-x) / z) + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+26], t$95$1, If[LessEqual[z, -4.1e-141], N[(N[(x * N[(y / z), $MachinePrecision] + N[(t - a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 6.6e-142], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 1.15], N[(N[(z / N[(b * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -3.29999999999999993e26

   1. Initial program 41.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 82.6

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

   4. Applied rewrites 82.6%

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

   if -3.29999999999999993e26 < z < -4.10000000000000002e-141

   1. Initial program 85.7%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{z}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x}{z} \]

      3. lower-/.f64 7.5

         \[\leadsto -\frac{x}{z} \]

   7. Applied rewrites 7.5%

      \[\leadsto -\frac{x}{z} \]

   8. Taylor expanded in b around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 41.8

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

   10. Applied rewrites 41.8%

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

   if -4.10000000000000002e-141 < z < 6.5999999999999994e-142

   1. Initial program 87.1%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 68.2%

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

   if 6.5999999999999994e-142 < z < 1.1499999999999999

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

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

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. div-add-rev N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-neg-in N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 53.3%

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

   if 1.1499999999999999 < z

   1. Initial program 46.4%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \frac{-1 \cdot x}{z} + \frac{\color{blue}{t} - a}{b - y} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.8

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

   7. Applied rewrites 84.8%

      \[\leadsto \frac{-x}{z} + \frac{\color{blue}{t} - a}{b - y} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{z}, t - a\right)}{b}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.3e+26)
     t_1
     (if (<= z -4.1e-141)
       (/ (fma x (/ y z) (- t a)) b)
       (if (<= z 4.4e-50) (/ (+ (* x y) (* z (- t a))) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.3e+26) {
		tmp = t_1;
	} else if (z <= -4.1e-141) {
		tmp = fma(x, (y / z), (t - a)) / b;
	} else if (z <= 4.4e-50) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.3e+26)
		tmp = t_1;
	elseif (z <= -4.1e-141)
		tmp = Float64(fma(x, Float64(y / z), Float64(t - a)) / b);
	elseif (z <= 4.4e-50)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+26], t$95$1, If[LessEqual[z, -4.1e-141], N[(N[(x * N[(y / z), $MachinePrecision] + N[(t - a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 4.4e-50], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.29999999999999993e26 or 4.3999999999999998e-50 < z

   1. Initial program 47.3%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 77.6

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

   4. Applied rewrites 77.6%

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

   if -3.29999999999999993e26 < z < -4.10000000000000002e-141

   1. Initial program 85.7%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{z}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x}{z} \]

      3. lower-/.f64 7.5

         \[\leadsto -\frac{x}{z} \]

   7. Applied rewrites 7.5%

      \[\leadsto -\frac{x}{z} \]

   8. Taylor expanded in b around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 41.8

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

   10. Applied rewrites 41.8%

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

   if -4.10000000000000002e-141 < z < 4.3999999999999998e-50

   1. Initial program 87.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 65.4%

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

Alternative 8: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{z}, t - a\right)}{b}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.3e+26)
     t_1
     (if (<= z -4.1e-141)
       (/ (fma x (/ y z) (- t a)) b)
       (if (<= z 5.6e-50) (fma (/ t y) z x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.3e+26) {
		tmp = t_1;
	} else if (z <= -4.1e-141) {
		tmp = fma(x, (y / z), (t - a)) / b;
	} else if (z <= 5.6e-50) {
		tmp = fma((t / y), z, 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 <= -3.3e+26)
		tmp = t_1;
	elseif (z <= -4.1e-141)
		tmp = Float64(fma(x, Float64(y / z), Float64(t - a)) / b);
	elseif (z <= 5.6e-50)
		tmp = fma(Float64(t / y), z, 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, -3.3e+26], t$95$1, If[LessEqual[z, -4.1e-141], N[(N[(x * N[(y / z), $MachinePrecision] + N[(t - a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 5.6e-50], N[(N[(t / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.29999999999999993e26 or 5.5999999999999996e-50 < z

   1. Initial program 47.3%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 77.6

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

   4. Applied rewrites 77.6%

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

   if -3.29999999999999993e26 < z < -4.10000000000000002e-141

   1. Initial program 85.7%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{z}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x}{z} \]

      3. lower-/.f64 7.5

         \[\leadsto -\frac{x}{z} \]

   7. Applied rewrites 7.5%

      \[\leadsto -\frac{x}{z} \]

   8. Taylor expanded in b around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 41.8

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

   10. Applied rewrites 41.8%

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

   if -4.10000000000000002e-141 < z < 5.5999999999999996e-50

   1. Initial program 87.2%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add-rev N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 52.4

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 60.5%

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

Alternative 9: 66.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -7.8e-35) t_1 (if (<= z 5.6e-50) (fma (/ t y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -7.8e-35) {
		tmp = t_1;
	} else if (z <= 5.6e-50) {
		tmp = fma((t / y), z, 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 <= -7.8e-35)
		tmp = t_1;
	elseif (z <= 5.6e-50)
		tmp = fma(Float64(t / y), z, 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, -7.8e-35], t$95$1, If[LessEqual[z, 5.6e-50], N[(N[(t / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.79999999999999961e-35 or 5.5999999999999996e-50 < z

   1. Initial program 50.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 75.1

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

   4. Applied rewrites 75.1%

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

   if -7.79999999999999961e-35 < z < 5.5999999999999996e-50

   1. Initial program 87.3%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add-rev N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 51.9

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

   4. Applied rewrites 51.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 58.2%

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

Alternative 10: 51.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+127}:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -6.5e+145)
     t_1
     (if (<= y 2.2e+84)
       (/ (- t a) b)
       (if (<= y 2.3e+127) (/ (- t a) (- y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -6.5e+145) {
		tmp = t_1;
	} else if (y <= 2.2e+84) {
		tmp = (t - a) / b;
	} else if (y <= 2.3e+127) {
		tmp = (t - a) / -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-6.5d+145)) then
        tmp = t_1
    else if (y <= 2.2d+84) then
        tmp = (t - a) / b
    else if (y <= 2.3d+127) then
        tmp = (t - a) / -y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -6.5e+145) {
		tmp = t_1;
	} else if (y <= 2.2e+84) {
		tmp = (t - a) / b;
	} else if (y <= 2.3e+127) {
		tmp = (t - a) / -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -6.5e+145:
		tmp = t_1
	elif y <= 2.2e+84:
		tmp = (t - a) / b
	elif y <= 2.3e+127:
		tmp = (t - a) / -y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -6.5e+145)
		tmp = t_1;
	elseif (y <= 2.2e+84)
		tmp = Float64(Float64(t - a) / b);
	elseif (y <= 2.3e+127)
		tmp = Float64(Float64(t - a) / Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -6.5e+145)
		tmp = t_1;
	elseif (y <= 2.2e+84)
		tmp = (t - a) / b;
	elseif (y <= 2.3e+127)
		tmp = (t - a) / -y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+145], t$95$1, If[LessEqual[y, 2.2e+84], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 2.3e+127], N[(N[(t - a), $MachinePrecision] / (-y)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.50000000000000034e145 or 2.3000000000000002e127 < y

   1. Initial program 43.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 64.1

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

   4. Applied rewrites 64.1%

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

   if -6.50000000000000034e145 < y < 2.1999999999999998e84

   1. Initial program 76.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 47.6

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

   4. Applied rewrites 47.6%

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

   if 2.1999999999999998e84 < y < 2.3000000000000002e127

   1. Initial program 59.5%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 40.5

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

   4. Applied rewrites 40.5%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{t - a}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 26.3

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

   7. Applied rewrites 26.3%

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

Alternative 11: 51.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -6.5e+145)
     t_1
     (if (<= y 4.1e+67)
       (/ (- t a) b)
       (if (<= y 1.6e+127) (/ t (- b y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -6.5e+145) {
		tmp = t_1;
	} else if (y <= 4.1e+67) {
		tmp = (t - a) / b;
	} else if (y <= 1.6e+127) {
		tmp = t / (b - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-6.5d+145)) then
        tmp = t_1
    else if (y <= 4.1d+67) then
        tmp = (t - a) / b
    else if (y <= 1.6d+127) then
        tmp = t / (b - y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -6.5e+145) {
		tmp = t_1;
	} else if (y <= 4.1e+67) {
		tmp = (t - a) / b;
	} else if (y <= 1.6e+127) {
		tmp = t / (b - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -6.5e+145:
		tmp = t_1
	elif y <= 4.1e+67:
		tmp = (t - a) / b
	elif y <= 1.6e+127:
		tmp = t / (b - y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -6.5e+145)
		tmp = t_1;
	elseif (y <= 4.1e+67)
		tmp = Float64(Float64(t - a) / b);
	elseif (y <= 1.6e+127)
		tmp = Float64(t / Float64(b - y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -6.5e+145)
		tmp = t_1;
	elseif (y <= 4.1e+67)
		tmp = (t - a) / b;
	elseif (y <= 1.6e+127)
		tmp = t / (b - y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+145], t$95$1, If[LessEqual[y, 4.1e+67], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 1.6e+127], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.50000000000000034e145 or 1.59999999999999988e127 < y

   1. Initial program 43.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 64.1

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

   4. Applied rewrites 64.1%

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

   if -6.50000000000000034e145 < y < 4.09999999999999979e67

   1. Initial program 77.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 48.2

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

   4. Applied rewrites 48.2%

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

   if 4.09999999999999979e67 < y < 1.59999999999999988e127

   1. Initial program 58.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 25.6%

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

Alternative 12: 45.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -7.5e+45) t_1 (if (<= z 5.6e-50) (/ x (- 1.0 z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -7.5e+45) {
		tmp = t_1;
	} else if (z <= 5.6e-50) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-7.5d+45)) then
        tmp = t_1
    else if (z <= 5.6d-50) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -7.5e+45) {
		tmp = t_1;
	} else if (z <= 5.6e-50) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -7.5e+45:
		tmp = t_1
	elif z <= 5.6e-50:
		tmp = x / (1.0 - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -7.5e+45)
		tmp = t_1;
	elseif (z <= 5.6e-50)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -7.5e+45)
		tmp = t_1;
	elseif (z <= 5.6e-50)
		tmp = x / (1.0 - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.5e+45], t$95$1, If[LessEqual[z, 5.6e-50], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000058e45 or 5.5999999999999996e-50 < z

   1. Initial program 46.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 43.1%

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

   if -7.50000000000000058e45 < z < 5.5999999999999996e-50

   1. Initial program 86.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 47.2

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

   4. Applied rewrites 47.2%

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

Alternative 13: 34.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 5.4e+52) (/ x (- 1.0 z)) (/ (- a) (- y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 5.4e+52) {
		tmp = x / (1.0 - z);
	} else {
		tmp = -a / -y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 5.4d+52) then
        tmp = x / (1.0d0 - z)
    else
        tmp = -a / -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 5.4e+52) {
		tmp = x / (1.0 - z);
	} else {
		tmp = -a / -y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= 5.4e+52:
		tmp = x / (1.0 - z)
	else:
		tmp = -a / -y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 5.4e+52)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(Float64(-a) / Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= 5.4e+52)
		tmp = x / (1.0 - z);
	else
		tmp = -a / -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 5.4e+52], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[((-a) / (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.4e52

   1. Initial program 67.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 35.7

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

   4. Applied rewrites 35.7%

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

   if 5.4e52 < a

   1. Initial program 63.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 57.3

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{t - a}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 25.6

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{-y} \]

      2. lower-neg.f64 20.6

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

   10. Applied rewrites 20.6%

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

Alternative 14: 34.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{-y}\\ \mathbf{if}\;z \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 45000:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- y))))
   (if (<= z -2e-15) t_1 (if (<= z 45000.0) (fma x z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / -y;
	double tmp;
	if (z <= -2e-15) {
		tmp = t_1;
	} else if (z <= 45000.0) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(-y))
	tmp = 0.0
	if (z <= -2e-15)
		tmp = t_1;
	elseif (z <= 45000.0)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / (-y)), $MachinePrecision]}, If[LessEqual[z, -2e-15], t$95$1, If[LessEqual[z, 45000.0], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000002e-15 or 45000 < z

   1. Initial program 46.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 79.2

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{t - a}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 21.3%

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

   if -2.0000000000000002e-15 < z < 45000

   1. Initial program 87.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. div-add-rev N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 50.3

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

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

Alternative 15: 33.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{-y}\\ \mathbf{if}\;z \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 45000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- y)))) (if (<= z -2e-15) t_1 (if (<= z 45000.0) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / -y;
	double tmp;
	if (z <= -2e-15) {
		tmp = t_1;
	} else if (z <= 45000.0) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / -y
    if (z <= (-2d-15)) then
        tmp = t_1
    else if (z <= 45000.0d0) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / -y;
	double tmp;
	if (z <= -2e-15) {
		tmp = t_1;
	} else if (z <= 45000.0) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / -y
	tmp = 0
	if z <= -2e-15:
		tmp = t_1
	elif z <= 45000.0:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(-y))
	tmp = 0.0
	if (z <= -2e-15)
		tmp = t_1;
	elseif (z <= 45000.0)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / -y;
	tmp = 0.0;
	if (z <= -2e-15)
		tmp = t_1;
	elseif (z <= 45000.0)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / (-y)), $MachinePrecision]}, If[LessEqual[z, -2e-15], t$95$1, If[LessEqual[z, 45000.0], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000002e-15 or 45000 < z

   1. Initial program 46.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 79.2

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{t - a}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 21.3%

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

   if -2.0000000000000002e-15 < z < 45000

   1. Initial program 87.7%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 47.2%

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

Alternative 16: 32.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (/ x z)))) (if (<= z -1.0) t_1 (if (<= z 1.18e-7) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 1.18e-7) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(x / z)
    if (z <= (-1.0d0)) then
        tmp = t_1
    else if (z <= 1.18d-7) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_1;
	} else if (z <= 1.18e-7) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -(x / z)
	tmp = 0
	if z <= -1.0:
		tmp = t_1
	elif z <= 1.18e-7:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(-Float64(x / z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 1.18e-7)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -(x / z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_1;
	elseif (z <= 1.18e-7)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = (-N[(x / z), $MachinePrecision])}, If[LessEqual[z, -1.0], t$95$1, If[LessEqual[z, 1.18e-7], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.18e-7 < z

   1. Initial program 46.1%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{z}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x}{z} \]

      3. lower-/.f64 18.8

         \[\leadsto -\frac{x}{z} \]

   7. Applied rewrites 18.8%

      \[\leadsto -\frac{x}{z} \]

   if -1 < z < 1.18e-7

   1. Initial program 87.4%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 47.4%

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

Alternative 17: 25.3% accurate, 23.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 66.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 25.3%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64
  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))