Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.9% → 94.7%

Time: 6.2s

Alternatives: 15

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% 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: 94.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, x \cdot \frac{y}{t\_2}\right)\\ t_4 := \left(b - y\right) \cdot \left(b - y\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+253}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-\frac{\left(-x \cdot \frac{y}{b - y}\right) - \left(-\frac{\left(t - a\right) \cdot y}{t\_4}\right)}{z}\right) + \frac{t - a}{b - y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{\left(b - y\right) \cdot z}, \frac{t}{b - y}\right) - \mathsf{fma}\left(y, \frac{t - a}{t\_4 \cdot z}, \frac{a}{b - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (fma (- b y) z y))
        (t_3 (fma z (/ (- t a) t_2) (* x (/ y t_2))))
        (t_4 (* (- b y) (- b y))))
   (if (<= t_1 -5e+253)
     t_3
     (if (<= t_1 -1e-306)
       t_1
       (if (<= t_1 0.0)
         (+
          (- (/ (- (- (* x (/ y (- b y)))) (- (/ (* (- t a) y) t_4))) z))
          (/ (- t a) (- b y)))
         (if (<= t_1 5e+290)
           t_1
           (if (<= t_1 INFINITY)
             t_3
             (-
              (fma x (/ y (* (- b y) z)) (/ t (- b y)))
              (fma y (/ (- t a) (* t_4 z)) (/ a (- b y)))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = fma((b - y), z, y);
	double t_3 = fma(z, ((t - a) / t_2), (x * (y / t_2)));
	double t_4 = (b - y) * (b - y);
	double tmp;
	if (t_1 <= -5e+253) {
		tmp = t_3;
	} else if (t_1 <= -1e-306) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = -((-(x * (y / (b - y))) - -(((t - a) * y) / t_4)) / z) + ((t - a) / (b - y));
	} else if (t_1 <= 5e+290) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = fma(x, (y / ((b - y) * z)), (t / (b - y))) - fma(y, ((t - a) / (t_4 * z)), (a / (b - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = fma(z, Float64(Float64(t - a) / t_2), Float64(x * Float64(y / t_2)))
	t_4 = Float64(Float64(b - y) * Float64(b - y))
	tmp = 0.0
	if (t_1 <= -5e+253)
		tmp = t_3;
	elseif (t_1 <= -1e-306)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(x * Float64(y / Float64(b - y)))) - Float64(-Float64(Float64(Float64(t - a) * y) / t_4))) / z)) + Float64(Float64(t - a) / Float64(b - y)));
	elseif (t_1 <= 5e+290)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(fma(x, Float64(y / Float64(Float64(b - y) * z)), Float64(t / Float64(b - y))) - fma(y, Float64(Float64(t - a) / Float64(t_4 * z)), Float64(a / Float64(b - y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(b - y), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+253], t$95$3, If[LessEqual[t$95$1, -1e-306], t$95$1, If[LessEqual[t$95$1, 0.0], N[((-N[(N[((-N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - (-N[(N[(N[(t - a), $MachinePrecision] * y), $MachinePrecision] / t$95$4), $MachinePrecision])), $MachinePrecision] / z), $MachinePrecision]) + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+290], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$3, N[(N[(x * N[(y / N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(N[(t - a), $MachinePrecision] / N[(t$95$4 * z), $MachinePrecision]), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

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)))) < -4.9999999999999997e253 or 4.9999999999999998e290 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 36.7%

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

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

      5. lift--.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 96.8%

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

   if -4.9999999999999997e253 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000003e-306 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.9999999999999998e290

   1. Initial program 99.4%

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

   if -1.00000000000000003e-306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 26.0%

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

   2. Taylor expanded in z around -inf

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

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

      3. lower-+.f64 N/A

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

   4. Applied rewrites 85.3%

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

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

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{t}{b - y} + \frac{x \cdot y}{z \cdot \left(b - y\right)}\right) - \left(\frac{a}{b - y} + \frac{y \cdot \left(t - a\right)}{z \cdot {\left(b - y\right)}^{2}}\right)} \]
   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. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 48.9%

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

Alternative 2: 94.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, x \cdot \frac{y}{t\_2}\right)\\ t_4 := \frac{t - a}{b - y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+253}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-\frac{\left(-x \cdot \frac{y}{b - y}\right) - \left(-\frac{\left(t - a\right) \cdot y}{\left(b - y\right) \cdot \left(b - y\right)}\right)}{z}\right) + t\_4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (fma (- b y) z y))
        (t_3 (fma z (/ (- t a) t_2) (* x (/ y t_2))))
        (t_4 (/ (- t a) (- b y))))
   (if (<= t_1 -5e+253)
     t_3
     (if (<= t_1 -1e-306)
       t_1
       (if (<= t_1 0.0)
         (+
          (-
           (/
            (-
             (- (* x (/ y (- b y))))
             (- (/ (* (- t a) y) (* (- b y) (- b y)))))
            z))
          t_4)
         (if (<= t_1 5e+290) t_1 (if (<= t_1 INFINITY) t_3 t_4)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = fma((b - y), z, y);
	double t_3 = fma(z, ((t - a) / t_2), (x * (y / t_2)));
	double t_4 = (t - a) / (b - y);
	double tmp;
	if (t_1 <= -5e+253) {
		tmp = t_3;
	} else if (t_1 <= -1e-306) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = -((-(x * (y / (b - y))) - -(((t - a) * y) / ((b - y) * (b - y)))) / z) + t_4;
	} else if (t_1 <= 5e+290) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = fma(z, Float64(Float64(t - a) / t_2), Float64(x * Float64(y / t_2)))
	t_4 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (t_1 <= -5e+253)
		tmp = t_3;
	elseif (t_1 <= -1e-306)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(x * Float64(y / Float64(b - y)))) - Float64(-Float64(Float64(Float64(t - a) * y) / Float64(Float64(b - y) * Float64(b - y))))) / z)) + t_4);
	elseif (t_1 <= 5e+290)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+253], t$95$3, If[LessEqual[t$95$1, -1e-306], t$95$1, If[LessEqual[t$95$1, 0.0], N[((-N[(N[((-N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - (-N[(N[(N[(t - a), $MachinePrecision] * y), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / z), $MachinePrecision]) + t$95$4), $MachinePrecision], If[LessEqual[t$95$1, 5e+290], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$3, t$95$4]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.9999999999999997e253 or 4.9999999999999998e290 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 36.7%

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

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

      5. lift--.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 96.8%

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

   if -4.9999999999999997e253 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000003e-306 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.9999999999999998e290

   1. Initial program 99.4%

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

   if -1.00000000000000003e-306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 26.0%

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

   2. Taylor expanded in z around -inf

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

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

      3. lower-+.f64 N/A

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

   4. Applied rewrites 85.3%

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

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

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 40.3

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

   4. Applied rewrites 40.3%

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

Alternative 3: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(b - y, z, y\right)\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_3}, x \cdot \frac{y}{t\_3}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+253}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma (- b y) z y))
        (t_4 (fma z (/ (- t a) t_3) (* x (/ y t_3)))))
   (if (<= t_2 -5e+253)
     t_4
     (if (<= t_2 -2e-311)
       t_2
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 5e+290) t_2 (if (<= t_2 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 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma((b - y), z, y);
	double t_4 = fma(z, ((t - a) / t_3), (x * (y / t_3)));
	double tmp;
	if (t_2 <= -5e+253) {
		tmp = t_4;
	} else if (t_2 <= -2e-311) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+290) {
		tmp = t_2;
	} else if (t_2 <= ((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 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(Float64(b - y), z, y)
	t_4 = fma(z, Float64(Float64(t - a) / t_3), Float64(x * Float64(y / t_3)))
	tmp = 0.0
	if (t_2 <= -5e+253)
		tmp = t_4;
	elseif (t_2 <= -2e-311)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e+290)
		tmp = t_2;
	elseif (t_2 <= 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[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(x * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+253], t$95$4, If[LessEqual[t$95$2, -2e-311], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 5e+290], t$95$2, If[LessEqual[t$95$2, 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)))) < -4.9999999999999997e253 or 4.9999999999999998e290 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 36.7%

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

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

      5. lift--.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 96.8%

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

   if -4.9999999999999997e253 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.9999999999999e-311 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.9999999999999998e290

   1. Initial program 99.4%

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

   if -1.9999999999999e-311 < (/.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 10.2%

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

   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 76.5

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

   4. Applied rewrites 76.5%

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

Alternative 4: 91.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -2e-311)
       t_2
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 5e+290) 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);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -2e-311) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+290) {
		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)), x)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -2e-311)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e+290)
		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] + x), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -2e-311], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 5e+290], 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 4.9999999999999998e290 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 32.0%

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

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

      5. lift--.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 99.1%

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

   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 82.7%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.9999999999999e-311 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.9999999999999998e290

   1. Initial program 99.4%

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

   if -1.9999999999999e-311 < (/.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 10.2%

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

   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 76.5

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

   4. Applied rewrites 76.5%

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

Alternative 5: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.55 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x \cdot \frac{y}{y}\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.8e-18)
     t_1
     (if (<= z -2.55e-144)
       (fma z (/ (- t a) y) (* x (/ y y)))
       (if (<= z 7.2e-34) (/ (+ (* x y) (* z (- t a))) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.8e-18) {
		tmp = t_1;
	} else if (z <= -2.55e-144) {
		tmp = fma(z, ((t - a) / y), (x * (y / y)));
	} else if (z <= 7.2e-34) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.8e-18)
		tmp = t_1;
	elseif (z <= -2.55e-144)
		tmp = fma(z, Float64(Float64(t - a) / y), Float64(x * Float64(y / y)));
	elseif (z <= 7.2e-34)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-18], t$95$1, If[LessEqual[z, -2.55e-144], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + N[(x * N[(y / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.2e-34], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.80000000000000012e-18 or 7.20000000000000016e-34 < z

   1. Initial program 50.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 76.6

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

   4. Applied rewrites 76.6%

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

   if -2.80000000000000012e-18 < z < -2.55e-144

   1. Initial program 87.8%

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

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

      5. lift--.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 94.8%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 62.0%

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

   if -2.55e-144 < z < 7.20000000000000016e-34

   1. Initial program 86.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 63.9%

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

Alternative 6: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 23000000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4e-16)
     t_1
     (if (<= z 23000000000000.0)
       (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 <= -4e-16) {
		tmp = t_1;
	} else if (z <= 23000000000000.0) {
		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 <= -4e-16)
		tmp = t_1;
	elseif (z <= 23000000000000.0)
		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, -4e-16], t$95$1, If[LessEqual[z, 23000000000000.0], 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 < -3.9999999999999999e-16 or 2.3e13 < 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 79.4

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

   4. Applied rewrites 79.4%

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

   if -3.9999999999999999e-16 < z < 2.3e13

   1. Initial program 87.1%

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

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

      5. lift--.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 90.4%

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

   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 73.7%

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

Alternative 7: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.8e-18)
     t_1
     (if (<= z 7.2e-34) (/ (+ (* x y) (* z (- t a))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.8e-18) {
		tmp = t_1;
	} else if (z <= 7.2e-34) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-2.8d-18)) then
        tmp = t_1
    else if (z <= 7.2d-34) then
        tmp = ((x * y) + (z * (t - a))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.8e-18) {
		tmp = t_1;
	} else if (z <= 7.2e-34) {
		tmp = ((x * y) + (z * (t - a))) / y;
	} 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 <= -2.8e-18:
		tmp = t_1
	elif z <= 7.2e-34:
		tmp = ((x * y) + (z * (t - a))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.8e-18)
		tmp = t_1;
	elseif (z <= 7.2e-34)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / y);
	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 <= -2.8e-18)
		tmp = t_1;
	elseif (z <= 7.2e-34)
		tmp = ((x * y) + (z * (t - a))) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-18], t$95$1, If[LessEqual[z, 7.2e-34], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.80000000000000012e-18 or 7.20000000000000016e-34 < z

   1. Initial program 50.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 76.6

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

   4. Applied rewrites 76.6%

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

   if -2.80000000000000012e-18 < z < 7.20000000000000016e-34

   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

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

   4. Applied rewrites 61.3%

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

Alternative 8: 68.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 29000000000000:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.6e-16)
     t_1
     (if (<= z 29000000000000.0) (* x (/ y (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 <= -3.6e-16) {
		tmp = t_1;
	} else if (z <= 29000000000000.0) {
		tmp = x * (y / 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 <= -3.6e-16)
		tmp = t_1;
	elseif (z <= 29000000000000.0)
		tmp = Float64(x * Float64(y / 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, -3.6e-16], t$95$1, If[LessEqual[z, 29000000000000.0], N[(x * N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999983e-16 or 2.9e13 < 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 79.4

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

   4. Applied rewrites 79.4%

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

   if -3.59999999999999983e-16 < z < 2.9e13

   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 x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 57.7

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

   4. Applied rewrites 57.7%

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

Alternative 9: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 22000000000000:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(b, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.6e-16)
     t_1
     (if (<= z 22000000000000.0) (* x (/ y (fma b 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 <= -3.6e-16) {
		tmp = t_1;
	} else if (z <= 22000000000000.0) {
		tmp = x * (y / fma(b, 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 <= -3.6e-16)
		tmp = t_1;
	elseif (z <= 22000000000000.0)
		tmp = Float64(x * Float64(y / fma(b, 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, -3.6e-16], t$95$1, If[LessEqual[z, 22000000000000.0], N[(x * N[(y / N[(b * z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999983e-16 or 2.2e13 < 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 79.4

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

   4. Applied rewrites 79.4%

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

   if -3.59999999999999983e-16 < z < 2.2e13

   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 x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 56.9%

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

Alternative 10: 64.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-x}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.05e-20) t_1 (if (<= z 6.2e-34) (/ (- x) (- z 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.05e-20) {
		tmp = t_1;
	} else if (z <= 6.2e-34) {
		tmp = -x / (z - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-2.05d-20)) then
        tmp = t_1
    else if (z <= 6.2d-34) then
        tmp = -x / (z - 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.05e-20) {
		tmp = t_1;
	} else if (z <= 6.2e-34) {
		tmp = -x / (z - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -2.05e-20:
		tmp = t_1
	elif z <= 6.2e-34:
		tmp = -x / (z - 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.05e-20)
		tmp = t_1;
	elseif (z <= 6.2e-34)
		tmp = Float64(Float64(-x) / Float64(z - 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -2.05e-20)
		tmp = t_1;
	elseif (z <= 6.2e-34)
		tmp = -x / (z - 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.05e-20], t$95$1, If[LessEqual[z, 6.2e-34], N[((-x) / N[(z - 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.05e-20 or 6.1999999999999996e-34 < z

   1. Initial program 50.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 76.5

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

   4. Applied rewrites 76.5%

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

   if -2.05e-20 < z < 6.1999999999999996e-34

   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 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 49.7

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

   4. Applied rewrites 49.7%

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

Alternative 11: 53.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-x}{z - 1}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- x) (- z 1.0))))
   (if (<= y -1.8e+15) t_1 (if (<= y 1.75e-6) (/ (- 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 <= -1.8e+15) {
		tmp = t_1;
	} else if (y <= 1.75e-6) {
		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 <= (-1.8d+15)) then
        tmp = t_1
    else if (y <= 1.75d-6) 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 <= -1.8e+15) {
		tmp = t_1;
	} else if (y <= 1.75e-6) {
		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 <= -1.8e+15:
		tmp = t_1
	elif y <= 1.75e-6:
		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 <= -1.8e+15)
		tmp = t_1;
	elseif (y <= 1.75e-6)
		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 <= -1.8e+15)
		tmp = t_1;
	elseif (y <= 1.75e-6)
		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, -1.8e+15], t$95$1, If[LessEqual[y, 1.75e-6], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e15 or 1.74999999999999997e-6 < y

   1. Initial program 53.7%

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

   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 50.9

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

   4. Applied rewrites 50.9%

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

   if -1.8e15 < y < 1.74999999999999997e-6

   1. Initial program 80.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 56.5

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

   4. Applied rewrites 56.5%

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

Alternative 12: 47.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) b)))
   (if (<= z -2.1e-20) t_1 (if (<= z 2.2e-26) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (z <= -2.1e-20) {
		tmp = t_1;
	} else if (z <= 2.2e-26) {
		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
    if (z <= (-2.1d-20)) then
        tmp = t_1
    else if (z <= 2.2d-26) 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;
	double tmp;
	if (z <= -2.1e-20) {
		tmp = t_1;
	} else if (z <= 2.2e-26) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / b
	tmp = 0
	if z <= -2.1e-20:
		tmp = t_1
	elif z <= 2.2e-26:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / b)
	tmp = 0.0
	if (z <= -2.1e-20)
		tmp = t_1;
	elseif (z <= 2.2e-26)
		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;
	tmp = 0.0;
	if (z <= -2.1e-20)
		tmp = t_1;
	elseif (z <= 2.2e-26)
		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] / b), $MachinePrecision]}, If[LessEqual[z, -2.1e-20], t$95$1, If[LessEqual[z, 2.2e-26], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0999999999999999e-20 or 2.2000000000000001e-26 < z

   1. Initial program 50.3%

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

   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 46.7

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

   4. Applied rewrites 46.7%

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

   if -2.0999999999999999e-20 < z < 2.2000000000000001e-26

   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

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

   4. Applied rewrites 49.3%

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

Alternative 13: 45.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 22000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -1.8e-16) t_1 (if (<= z 22000000000000.0) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -1.8e-16) {
		tmp = t_1;
	} else if (z <= 22000000000000.0) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-1.8d-16)) then
        tmp = t_1
    else if (z <= 22000000000000.0d0) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -1.8e-16) {
		tmp = t_1;
	} else if (z <= 22000000000000.0) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -1.8e-16:
		tmp = t_1
	elif z <= 22000000000000.0:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -1.8e-16)
		tmp = t_1;
	elseif (z <= 22000000000000.0)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -1.8e-16)
		tmp = t_1;
	elseif (z <= 22000000000000.0)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e-16], t$95$1, If[LessEqual[z, 22000000000000.0], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.79999999999999991e-16 or 2.2e13 < 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 a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 30.3

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 43.7

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

   7. Applied rewrites 43.7%

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

   if -1.79999999999999991e-16 < z < 2.2e13

   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 46.8%

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

Alternative 14: 36.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 22000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.8e-16) (/ t b) (if (<= z 22000000000000.0) x (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.8e-16) {
		tmp = t / b;
	} else if (z <= 22000000000000.0) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-1.8d-16)) then
        tmp = t / b
    else if (z <= 22000000000000.0d0) then
        tmp = x
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.8e-16) {
		tmp = t / b;
	} else if (z <= 22000000000000.0) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.8e-16:
		tmp = t / b
	elif z <= 22000000000000.0:
		tmp = x
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.8e-16)
		tmp = Float64(t / b);
	elseif (z <= 22000000000000.0)
		tmp = x;
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.8e-16)
		tmp = t / b;
	elseif (z <= 22000000000000.0)
		tmp = x;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.8e-16], N[(t / b), $MachinePrecision], If[LessEqual[z, 22000000000000.0], x, N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.79999999999999991e-16 or 2.2e13 < 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 a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 30.3

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 27.0

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

   7. Applied rewrites 27.0%

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

   if -1.79999999999999991e-16 < z < 2.2e13

   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 46.8%

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

Alternative 15: 24.9% accurate, 23.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 24.9%

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

Reproduce

herbie shell --seed 2025115 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64
  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))