Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 65.5% → 96.2%

Time: 5.9s

Alternatives: 17

Speedup: 1.3×

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: 65.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{t - a}{{\left(b - y\right)}^{2}}, \frac{a}{b - y}\right)\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, x \cdot \frac{y}{t\_2}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ x z) (/ y (- b y)) (/ t (- b y)))
          (fma (/ y z) (/ (- t a) (pow (- b y) 2.0)) (/ a (- b y)))))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma z (/ (- t a) t_2) (* x (/ y t_2)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -2e-304)
       t_3
       (if (<= t_3 0.0) t_1 (if (<= t_3 INFINITY) t_4 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((x / z), (y / (b - y)), (t / (b - y))) - fma((y / z), ((t - a) / pow((b - y), 2.0)), (a / (b - y)));
	double t_2 = fma((b - y), z, y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma(z, ((t - a) / t_2), (x * (y / t_2)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -2e-304) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(x / z), Float64(y / Float64(b - y)), Float64(t / Float64(b - y))) - fma(Float64(y / z), Float64(Float64(t - a) / (Float64(b - y) ^ 2.0)), Float64(a / Float64(b - y))))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(z, Float64(Float64(t - a) / t_2), Float64(x * Float64(y / t_2)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -2e-304)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x / z), $MachinePrecision] * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / z), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -2e-304], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 93.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999994e-304

   1. Initial program 99.4%

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

   if -1.99999999999999994e-304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 10.9%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 98.6%

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

Alternative 2: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_1}, x \cdot \frac{y}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-x \cdot \frac{y}{b - y}\right) - \left(-\frac{\left(t - a\right) \cdot y}{{\left(b - y\right)}^{2}}\right)}{z}, -1, t\_2\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma z (/ (- t a) t_1) (* x (/ y t_1)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -2e-304)
       t_3
       (if (<= t_3 0.0)
         (fma
          (/
           (- (- (* x (/ y (- b y)))) (- (/ (* (- t a) y) (pow (- b y) 2.0))))
           z)
          -1.0
          t_2)
         (if (<= t_3 INFINITY) t_4 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 t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma(z, ((t - a) / t_1), (x * (y / t_1)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -2e-304) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = fma(((-(x * (y / (b - y))) - -(((t - a) * y) / pow((b - y), 2.0))) / z), -1.0, t_2);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(z, Float64(Float64(t - a) / t_1), Float64(x * Float64(y / t_1)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -2e-304)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = fma(Float64(Float64(Float64(-Float64(x * Float64(y / Float64(b - y)))) - Float64(-Float64(Float64(Float64(t - a) * y) / (Float64(b - y) ^ 2.0)))) / z), -1.0, t_2);
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -2e-304], t$95$3, If[LessEqual[t$95$3, 0.0], N[(N[(N[((-N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - (-N[(N[(N[(t - a), $MachinePrecision] * y), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / z), $MachinePrecision] * -1.0 + t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 93.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999994e-304

   1. Initial program 99.4%

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

   if -1.99999999999999994e-304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 28.9%

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

   2. Taylor expanded in z around -inf

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. div-sub N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 86.0%

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

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

   1. Initial program 0.0%

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

   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 73.3

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

   4. Applied rewrites 73.3%

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

Alternative 3: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x \cdot \frac{-1}{z - 1}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 10^{-291}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+125}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (/ (- t a) (fma (- b y) z y)) (* x (/ -1.0 (- z 1.0))))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -2e-304)
       t_2
       (if (<= t_2 1e-291)
         t_1
         (if (<= t_2 5e+125) t_2 (if (<= t_2 INFINITY) t_3 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, ((t - a) / fma((b - y), z, y)), (x * (-1.0 / (z - 1.0))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -2e-304) {
		tmp = t_2;
	} else if (t_2 <= 1e-291) {
		tmp = t_1;
	} else if (t_2 <= 5e+125) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), Float64(x * Float64(-1.0 / Float64(z - 1.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -2e-304)
		tmp = t_2;
	elseif (t_2 <= 1e-291)
		tmp = t_1;
	elseif (t_2 <= 5e+125)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	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]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -2e-304], t$95$2, If[LessEqual[t$95$2, 1e-291], t$95$1, If[LessEqual[t$95$2, 5e+125], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 4.99999999999999962e125 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 49.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 92.2%

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

   4. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.3

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

   6. Applied rewrites 87.3%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999994e-304 or 9.99999999999999962e-292 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.99999999999999962e125

   1. Initial program 99.4%

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

   if -1.99999999999999994e-304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.99999999999999962e-292 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 11.8%

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

   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 76.0

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

   4. Applied rewrites 76.0%

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

Alternative 4: 87.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := z \cdot \left(b - y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{t\_2}, x\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 10^{-291}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+125}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (* z (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y t_2))))
   (if (<= t_3 (- INFINITY))
     (fma z (/ (- t a) t_2) x)
     (if (<= t_3 -2e-304)
       t_3
       (if (<= t_3 1e-291)
         t_1
         (if (<= t_3 5e+125)
           t_3
           (if (<= t_3 INFINITY)
             (fma z (/ (- t a) (fma (- b y) z y)) x)
             t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = z * (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + t_2);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(z, ((t - a) / t_2), x);
	} else if (t_3 <= -2e-304) {
		tmp = t_3;
	} else if (t_3 <= 1e-291) {
		tmp = t_1;
	} else if (t_3 <= 5e+125) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma(z, ((t - a) / fma((b - y), z, y)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(z * Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + t_2))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(z, Float64(Float64(t - a) / t_2), x);
	elseif (t_3 <= -2e-304)
		tmp = t_3;
	elseif (t_3 <= 1e-291)
		tmp = t_1;
	elseif (t_3 <= 5e+125)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$3, -2e-304], t$95$3, If[LessEqual[t$95$3, 1e-291], t$95$1, If[LessEqual[t$95$3, 5e+125], t$95$3, If[LessEqual[t$95$3, Infinity], N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lift--.f64 79.4

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

   9. Applied rewrites 79.4%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999994e-304 or 9.99999999999999962e-292 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.99999999999999962e125

   1. Initial program 99.4%

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

   if -1.99999999999999994e-304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.99999999999999962e-292 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 11.8%

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

   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 76.0

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

   4. Applied rewrites 76.0%

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

   if 4.99999999999999962e125 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 61.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 87.7%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 73.4%

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

Alternative 5: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, x \cdot \frac{y}{t\_2}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma z (/ (- t a) t_2) (* x (/ y t_2)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -2e-304)
       t_3
       (if (<= t_3 0.0) t_1 (if (<= t_3 INFINITY) t_4 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = fma((b - y), z, y);
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma(z, ((t - a) / t_2), (x * (y / t_2)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -2e-304) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(z, Float64(Float64(t - a) / t_2), Float64(x * Float64(y / t_2)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -2e-304)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= Inf)
		tmp = t_4;
	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]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -2e-304], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 93.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999994e-304

   1. Initial program 99.4%

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

   if -1.99999999999999994e-304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 10.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   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 76.1

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

   4. Applied rewrites 76.1%

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

Alternative 6: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.027:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{t - a}{y} \cdot z + x\\ \mathbf{elif}\;z \leq 9000000000000:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.027)
     t_1
     (if (<= z 4.5e-47)
       (+ (* (/ (- t a) y) z) x)
       (if (<= z 9000000000000.0) (/ (* (- t a) z) (fma (- b y) z 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 <= -0.027) {
		tmp = t_1;
	} else if (z <= 4.5e-47) {
		tmp = (((t - a) / y) * z) + x;
	} else if (z <= 9000000000000.0) {
		tmp = ((t - a) * z) / fma((b - y), z, 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 <= -0.027)
		tmp = t_1;
	elseif (z <= 4.5e-47)
		tmp = Float64(Float64(Float64(Float64(t - a) / y) * z) + x);
	elseif (z <= 9000000000000.0)
		tmp = Float64(Float64(Float64(t - a) * z) / fma(Float64(b - y), z, 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, -0.027], t$95$1, If[LessEqual[z, 4.5e-47], N[(N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 9000000000000.0], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.0269999999999999997 or 9e12 < z

   1. Initial program 44.1%

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

   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 79.5

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

   4. Applied rewrites 79.5%

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

   if -0.0269999999999999997 < z < 4.5e-47

   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. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.6%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 64.6

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

   9. Applied rewrites 64.6%

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

   if 4.5e-47 < z < 9e12

   1. Initial program 88.7%

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

   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 63.6

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

   4. Applied rewrites 63.6%

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

Alternative 7: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.3e+21)
     t_1
     (if (<= z 5000000.0) (fma z (/ (- t a) (fma (- b y) z y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.3e+21) {
		tmp = t_1;
	} else if (z <= 5000000.0) {
		tmp = fma(z, ((t - a) / fma((b - y), z, y)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.3e+21)
		tmp = t_1;
	elseif (z <= 5000000.0)
		tmp = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+21], t$95$1, If[LessEqual[z, 5000000.0], N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e21 or 5e6 < z

   1. Initial program 42.8%

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

   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 80.4

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

   4. Applied rewrites 80.4%

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

   if -3.3e21 < z < 5e6

   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

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 90.2%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 72.7%

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

Alternative 8: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.027:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.011:\\ \;\;\;\;\frac{t - a}{y} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.027) t_1 (if (<= z 0.011) (+ (* (/ (- t a) 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 <= -0.027) {
		tmp = t_1;
	} else if (z <= 0.011) {
		tmp = (((t - a) / y) * z) + 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 - a) / (b - y)
    if (z <= (-0.027d0)) then
        tmp = t_1
    else if (z <= 0.011d0) then
        tmp = (((t - a) / y) * z) + 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 - a) / (b - y);
	double tmp;
	if (z <= -0.027) {
		tmp = t_1;
	} else if (z <= 0.011) {
		tmp = (((t - a) / y) * z) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -0.027:
		tmp = t_1
	elif z <= 0.011:
		tmp = (((t - a) / 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 <= -0.027)
		tmp = t_1;
	elseif (z <= 0.011)
		tmp = Float64(Float64(Float64(Float64(t - a) / y) * z) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -0.027)
		tmp = t_1;
	elseif (z <= 0.011)
		tmp = (((t - a) / y) * z) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.027], t$95$1, If[LessEqual[z, 0.011], N[(N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0269999999999999997 or 0.010999999999999999 < z

   1. Initial program 45.0%

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

   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 78.9

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

   4. Applied rewrites 78.9%

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

   if -0.0269999999999999997 < z < 0.010999999999999999

   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. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.7%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 63.7

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

   9. Applied rewrites 63.7%

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

Alternative 9: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.027:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.027) t_1 (if (<= z 0.011) (fma (/ (- t a) 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 <= -0.027) {
		tmp = t_1;
	} else if (z <= 0.011) {
		tmp = fma(((t - a) / 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 <= -0.027)
		tmp = t_1;
	elseif (z <= 0.011)
		tmp = fma(Float64(Float64(t - a) / 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, -0.027], t$95$1, If[LessEqual[z, 0.011], N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0269999999999999997 or 0.010999999999999999 < z

   1. Initial program 45.0%

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

   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 78.9

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

   4. Applied rewrites 78.9%

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

   if -0.0269999999999999997 < z < 0.010999999999999999

   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. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.7%

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

Alternative 10: 34.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+266}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq -0.35:\\ \;\;\;\;\frac{a}{y}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.25e+266)
   (/ t b)
   (if (<= z -0.35) (/ a y) (if (<= z 1.4e-58) (fma x z x) (/ t b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.25e+266) {
		tmp = t / b;
	} else if (z <= -0.35) {
		tmp = a / y;
	} else if (z <= 1.4e-58) {
		tmp = fma(x, z, x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.25e+266)
		tmp = Float64(t / b);
	elseif (z <= -0.35)
		tmp = Float64(a / y);
	elseif (z <= 1.4e-58)
		tmp = fma(x, z, x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.25e+266], N[(t / b), $MachinePrecision], If[LessEqual[z, -0.35], N[(a / y), $MachinePrecision], If[LessEqual[z, 1.4e-58], N[(x * z + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e266 or 1.4e-58 < z

   1. Initial program 47.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 6.6%

      \[\leadsto \color{blue}{x} \]

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-*.f64 26.8

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

   7. Applied rewrites 26.8%

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

   8. Taylor expanded in t around inf

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

      1. lower-/.f64 25.9

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

   10. Applied rewrites 25.9%

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

   if -1.25e266 < z < -0.34999999999999998

   1. Initial program 50.1%

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

   2. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 22.5

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

   4. Applied rewrites 22.5%

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

   5. Taylor expanded in y around -inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 19.3

         \[\leadsto \frac{a}{y} \cdot \frac{z}{z - 1} \]

   7. Applied rewrites 19.3%

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

   8. Taylor expanded in z around inf

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

      1. lift-/.f64 19.1

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

   10. Applied rewrites 19.1%

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

   if -0.34999999999999998 < z < 1.4e-58

   1. Initial program 87.0%

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

   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. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 49.0%

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

Alternative 11: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.027:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.027) t_1 (if (<= z 2.7e-40) (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 <= -0.027) {
		tmp = t_1;
	} else if (z <= 2.7e-40) {
		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 <= -0.027)
		tmp = t_1;
	elseif (z <= 2.7e-40)
		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, -0.027], t$95$1, If[LessEqual[z, 2.7e-40], N[(N[(t / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0269999999999999997 or 2.7e-40 < z

   1. Initial program 47.5%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

   if -0.0269999999999999997 < z < 2.7e-40

   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. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 50.6

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

   4. Applied rewrites 50.6%

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

   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 56.1%

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

Alternative 12: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{-122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.8e-122) t_1 (if (<= z 1.7e-59) 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 <= -4.8e-122) {
		tmp = t_1;
	} else if (z <= 1.7e-59) {
		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 - a) / (b - y)
    if (z <= (-4.8d-122)) then
        tmp = t_1
    else if (z <= 1.7d-59) 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 - a) / (b - y);
	double tmp;
	if (z <= -4.8e-122) {
		tmp = t_1;
	} else if (z <= 1.7e-59) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -4.8e-122:
		tmp = t_1
	elif z <= 1.7e-59:
		tmp = 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 <= -4.8e-122)
		tmp = t_1;
	elseif (z <= 1.7e-59)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -4.8e-122)
		tmp = t_1;
	elseif (z <= 1.7e-59)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.8e-122], t$95$1, If[LessEqual[z, 1.7e-59], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.79999999999999975e-122 or 1.70000000000000009e-59 < z

   1. Initial program 54.2%

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

   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 69.6

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

   4. Applied rewrites 69.6%

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

   if -4.79999999999999975e-122 < z < 1.70000000000000009e-59

   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

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

   4. Applied rewrites 53.5%

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

Alternative 13: 53.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-x}{z - 1}\\ \mathbf{if}\;y \leq -460000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- x) (- z 1.0))))
   (if (<= y -460000000.0) t_1 (if (<= y 1.8e+45) (/ (- t a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -x / (z - 1.0);
	double tmp;
	if (y <= -460000000.0) {
		tmp = t_1;
	} else if (y <= 1.8e+45) {
		tmp = (t - a) / b;
	} 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 - 1.0d0)
    if (y <= (-460000000.0d0)) then
        tmp = t_1
    else if (y <= 1.8d+45) then
        tmp = (t - a) / b
    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 - 1.0);
	double tmp;
	if (y <= -460000000.0) {
		tmp = t_1;
	} else if (y <= 1.8e+45) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -x / (z - 1.0)
	tmp = 0
	if y <= -460000000.0:
		tmp = t_1
	elif y <= 1.8e+45:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-x) / Float64(z - 1.0))
	tmp = 0.0
	if (y <= -460000000.0)
		tmp = t_1;
	elseif (y <= 1.8e+45)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -x / (z - 1.0);
	tmp = 0.0;
	if (y <= -460000000.0)
		tmp = t_1;
	elseif (y <= 1.8e+45)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-x) / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -460000000.0], t$95$1, If[LessEqual[y, 1.8e+45], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e8 or 1.8e45 < y

   1. Initial program 49.5%

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

   2. Taylor expanded in y around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 53.6

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

   4. Applied rewrites 53.6%

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

   if -4.6e8 < y < 1.8e45

   1. Initial program 79.2%

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

   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 54.2

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

   4. Applied rewrites 54.2%

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

Alternative 14: 46.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.1e+18) x (if (<= y 9.2e+16) (/ (- t a) b) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.1e+18) {
		tmp = x;
	} else if (y <= 9.2e+16) {
		tmp = (t - a) / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.1d+18)) then
        tmp = x
    else if (y <= 9.2d+16) then
        tmp = (t - a) / b
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.1e+18) {
		tmp = x;
	} else if (y <= 9.2e+16) {
		tmp = (t - a) / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.1e+18:
		tmp = x
	elif y <= 9.2e+16:
		tmp = (t - a) / b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.1e+18)
		tmp = x;
	elseif (y <= 9.2e+16)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.1e+18)
		tmp = x;
	elseif (y <= 9.2e+16)
		tmp = (t - a) / b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.1e+18], x, If[LessEqual[y, 9.2e+16], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1e18 or 9.2e16 < y

   1. Initial program 50.0%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 36.9%

      \[\leadsto \color{blue}{x} \]

   if -2.1e18 < y < 9.2e16

   1. Initial program 79.6%

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

   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 55.0

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

   4. Applied rewrites 55.0%

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

Alternative 15: 33.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.35:\\ \;\;\;\;\frac{a}{y}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -0.35) (/ a y) (if (<= z 1.75e-34) (fma x z x) (/ a y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -0.35) {
		tmp = a / y;
	} else if (z <= 1.75e-34) {
		tmp = fma(x, z, x);
	} else {
		tmp = a / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -0.35)
		tmp = Float64(a / y);
	elseif (z <= 1.75e-34)
		tmp = fma(x, z, x);
	else
		tmp = Float64(a / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -0.35], N[(a / y), $MachinePrecision], If[LessEqual[z, 1.75e-34], N[(x * z + x), $MachinePrecision], N[(a / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.34999999999999998 or 1.75e-34 < z

   1. Initial program 47.2%

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

   2. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 22.5

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

   4. Applied rewrites 22.5%

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

   5. Taylor expanded in y around -inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 21.3

         \[\leadsto \frac{a}{y} \cdot \frac{z}{z - 1} \]

   7. Applied rewrites 21.3%

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

   8. Taylor expanded in z around inf

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

      1. lift-/.f64 20.4

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

   10. Applied rewrites 20.4%

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

   if -0.34999999999999998 < z < 1.75e-34

   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. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 48.2%

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

Alternative 16: 24.8% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

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. div-add-rev N/A

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

   5. sub-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift--.f64 27.4

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

4. Applied rewrites 27.4%

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

5. Taylor expanded in y around inf

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

7. Applied rewrites 24.8%

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

Alternative 17: 24.6% accurate, 39.0× 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 65.5%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 24.6%

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

Developer Target 1: 72.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z)))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))