Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B

Percentage Accurate: 88.9% → 97.9%

Time: 7.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)} \leq 0:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left(z - t\right) - y, z, t \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (/ x_m (* (- y z) (- t z))) 0.0)
    (/ (/ x_m (- t z)) (- y z))
    (/ x_m (fma (- (- z t) y) z (* t y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m / ((y - z) * (t - z))) <= 0.0) {
		tmp = (x_m / (t - z)) / (y - z);
	} else {
		tmp = x_m / fma(((z - t) - y), z, (t * y));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m / Float64(Float64(y - z) * Float64(t - z))) <= 0.0)
		tmp = Float64(Float64(x_m / Float64(t - z)) / Float64(y - z));
	else
		tmp = Float64(x_m / fma(Float64(Float64(z - t) - y), z, Float64(t * y)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(z - t), $MachinePrecision] - y), $MachinePrecision] * z + N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < 0.0

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 0.0 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 2: 92.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left(z - t\right) - y, z, t \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\left(-t\right) + z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (/ (/ x_m (- y z)) t)
      (if (<= t_1 4e+288)
        (/ x_m (fma (- (- z t) y) z (* t y)))
        (/ (/ x_m z) (+ (- t) z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x_m / (y - z)) / t;
	} else if (t_1 <= 4e+288) {
		tmp = x_m / fma(((z - t) - y), z, (t * y));
	} else {
		tmp = (x_m / z) / (-t + z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(y - z) * Float64(t - z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x_m / Float64(y - z)) / t);
	elseif (t_1 <= 4e+288)
		tmp = Float64(x_m / fma(Float64(Float64(z - t) - y), z, Float64(t * y)));
	else
		tmp = Float64(Float64(x_m / z) / Float64(Float64(-t) + z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 4e+288], N[(x$95$m / N[(N[(N[(z - t), $MachinePrecision] - y), $MachinePrecision] * z + N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / N[((-t) + z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 y z) (-.f64 t z)) < -inf.0

   1. Initial program 69.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -inf.0 < (*.f64 (-.f64 y z) (-.f64 t z)) < 4e288

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 4e288 < (*.f64 (-.f64 y z) (-.f64 t z))

   1. Initial program 79.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{elif}\;\left(y - z\right) \cdot \left(t - z\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(z - t\right) - y, z, t \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\left(-t\right) + z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x\_m}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\left(-t\right) + z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (/ (/ x_m (- y z)) t)
      (if (<= t_1 4e+288) (/ x_m t_1) (/ (/ x_m z) (+ (- t) z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x_m / (y - z)) / t;
	} else if (t_1 <= 4e+288) {
		tmp = x_m / t_1;
	} else {
		tmp = (x_m / z) / (-t + z);
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x_m / (y - z)) / t;
	} else if (t_1 <= 4e+288) {
		tmp = x_m / t_1;
	} else {
		tmp = (x_m / z) / (-t + z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (y - z) * (t - z)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x_m / (y - z)) / t
	elif t_1 <= 4e+288:
		tmp = x_m / t_1
	else:
		tmp = (x_m / z) / (-t + z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(y - z) * Float64(t - z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x_m / Float64(y - z)) / t);
	elseif (t_1 <= 4e+288)
		tmp = Float64(x_m / t_1);
	else
		tmp = Float64(Float64(x_m / z) / Float64(Float64(-t) + z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (y - z) * (t - z);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x_m / (y - z)) / t;
	elseif (t_1 <= 4e+288)
		tmp = x_m / t_1;
	else
		tmp = (x_m / z) / (-t + z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 4e+288], N[(x$95$m / t$95$1), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / N[((-t) + z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 y z) (-.f64 t z)) < -inf.0

   1. Initial program 69.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -inf.0 < (*.f64 (-.f64 y z) (-.f64 t z)) < 4e288

   1. Initial program 99.1%

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

   if 4e288 < (*.f64 (-.f64 y z) (-.f64 t z))

   1. Initial program 79.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{elif}\;\left(y - z\right) \cdot \left(t - z\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\left(-t\right) + z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot y}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x\_m}{\left(z - y\right) \cdot z}\\ \mathbf{elif}\;t \leq 130000:\\ \;\;\;\;\frac{x\_m}{\left(z - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -5.2e-103)
    (/ x_m (* (- t z) y))
    (if (<= t 3.4e-75)
      (/ x_m (* (- z y) z))
      (if (<= t 130000.0) (/ x_m (* (- z t) z)) (/ x_m (* (- y z) t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -5.2e-103) {
		tmp = x_m / ((t - z) * y);
	} else if (t <= 3.4e-75) {
		tmp = x_m / ((z - y) * z);
	} else if (t <= 130000.0) {
		tmp = x_m / ((z - t) * z);
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.2d-103)) then
        tmp = x_m / ((t - z) * y)
    else if (t <= 3.4d-75) then
        tmp = x_m / ((z - y) * z)
    else if (t <= 130000.0d0) then
        tmp = x_m / ((z - t) * z)
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -5.2e-103) {
		tmp = x_m / ((t - z) * y);
	} else if (t <= 3.4e-75) {
		tmp = x_m / ((z - y) * z);
	} else if (t <= 130000.0) {
		tmp = x_m / ((z - t) * z);
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -5.2e-103:
		tmp = x_m / ((t - z) * y)
	elif t <= 3.4e-75:
		tmp = x_m / ((z - y) * z)
	elif t <= 130000.0:
		tmp = x_m / ((z - t) * z)
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -5.2e-103)
		tmp = Float64(x_m / Float64(Float64(t - z) * y));
	elseif (t <= 3.4e-75)
		tmp = Float64(x_m / Float64(Float64(z - y) * z));
	elseif (t <= 130000.0)
		tmp = Float64(x_m / Float64(Float64(z - t) * z));
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -5.2e-103)
		tmp = x_m / ((t - z) * y);
	elseif (t <= 3.4e-75)
		tmp = x_m / ((z - y) * z);
	elseif (t <= 130000.0)
		tmp = x_m / ((z - t) * z);
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -5.2e-103], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-75], N[(x$95$m / N[(N[(z - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 130000.0], N[(x$95$m / N[(N[(z - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < -5.19999999999999993e-103

   1. Initial program 90.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 51.3

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

   5. Applied rewrites 51.3%

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

   if -5.19999999999999993e-103 < t < 3.40000000000000015e-75

   1. Initial program 91.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.7%

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

   if 3.40000000000000015e-75 < t < 1.3e5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.3%

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

   if 1.3e5 < t

   1. Initial program 89.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 81.4

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

   5. Applied rewrites 81.4%

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

Alternative 5: 60.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -75000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{x\_m}{\left(-z\right) \cdot y}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z z))))
   (*
    x_s
    (if (<= z -75000000000000.0)
      t_1
      (if (<= z -9.2e-139)
        (/ x_m (* (- z) y))
        (if (<= z 1.12e-11) (/ x_m (* t y)) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -75000000000000.0) {
		tmp = t_1;
	} else if (z <= -9.2e-139) {
		tmp = x_m / (-z * y);
	} else if (z <= 1.12e-11) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * z)
    if (z <= (-75000000000000.0d0)) then
        tmp = t_1
    else if (z <= (-9.2d-139)) then
        tmp = x_m / (-z * y)
    else if (z <= 1.12d-11) then
        tmp = x_m / (t * y)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -75000000000000.0) {
		tmp = t_1;
	} else if (z <= -9.2e-139) {
		tmp = x_m / (-z * y);
	} else if (z <= 1.12e-11) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -75000000000000.0:
		tmp = t_1
	elif z <= -9.2e-139:
		tmp = x_m / (-z * y)
	elif z <= 1.12e-11:
		tmp = x_m / (t * y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -75000000000000.0)
		tmp = t_1;
	elseif (z <= -9.2e-139)
		tmp = Float64(x_m / Float64(Float64(-z) * y));
	elseif (z <= 1.12e-11)
		tmp = Float64(x_m / Float64(t * y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -75000000000000.0)
		tmp = t_1;
	elseif (z <= -9.2e-139)
		tmp = x_m / (-z * y);
	elseif (z <= 1.12e-11)
		tmp = x_m / (t * y);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -75000000000000.0], t$95$1, If[LessEqual[z, -9.2e-139], N[(x$95$m / N[((-z) * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.12e-11], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5e13 or 1.1200000000000001e-11 < z

   1. Initial program 85.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -7.5e13 < z < -9.2000000000000005e-139

   1. Initial program 96.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 44.4%

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

   if -9.2000000000000005e-139 < z < 1.1200000000000001e-11

   1. Initial program 97.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 65.9

         \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]

   5. Applied rewrites 65.9%

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

Alternative 6: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+35} \lor \neg \left(y \leq 1.72 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(z - t\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -8.2e+35) (not (<= y 1.72e-44)))
    (/ x_m (* (- t z) y))
    (/ x_m (* (- z t) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -8.2e+35) || !(y <= 1.72e-44)) {
		tmp = x_m / ((t - z) * y);
	} else {
		tmp = x_m / ((z - t) * z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-8.2d+35)) .or. (.not. (y <= 1.72d-44))) then
        tmp = x_m / ((t - z) * y)
    else
        tmp = x_m / ((z - t) * z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -8.2e+35) || !(y <= 1.72e-44)) {
		tmp = x_m / ((t - z) * y);
	} else {
		tmp = x_m / ((z - t) * z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -8.2e+35) or not (y <= 1.72e-44):
		tmp = x_m / ((t - z) * y)
	else:
		tmp = x_m / ((z - t) * z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -8.2e+35) || !(y <= 1.72e-44))
		tmp = Float64(x_m / Float64(Float64(t - z) * y));
	else
		tmp = Float64(x_m / Float64(Float64(z - t) * z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -8.2e+35) || ~((y <= 1.72e-44)))
		tmp = x_m / ((t - z) * y);
	else
		tmp = x_m / ((z - t) * z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -8.2e+35], N[Not[LessEqual[y, 1.72e-44]], $MachinePrecision]], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(z - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -8.1999999999999997e35 or 1.72e-44 < y

   1. Initial program 91.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if -8.1999999999999997e35 < y < 1.72e-44

   1. Initial program 90.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+35} \lor \neg \left(y \leq 1.72 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-108} \lor \neg \left(z \leq 2.05 \cdot 10^{-199}\right):\\ \;\;\;\;\frac{x\_m}{\left(z - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -3.1e-108) (not (<= z 2.05e-199)))
    (/ x_m (* (- z t) z))
    (/ x_m (* t y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -3.1e-108) || !(z <= 2.05e-199)) {
		tmp = x_m / ((z - t) * z);
	} else {
		tmp = x_m / (t * y);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-3.1d-108)) .or. (.not. (z <= 2.05d-199))) then
        tmp = x_m / ((z - t) * z)
    else
        tmp = x_m / (t * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -3.1e-108) || !(z <= 2.05e-199)) {
		tmp = x_m / ((z - t) * z);
	} else {
		tmp = x_m / (t * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -3.1e-108) or not (z <= 2.05e-199):
		tmp = x_m / ((z - t) * z)
	else:
		tmp = x_m / (t * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -3.1e-108) || !(z <= 2.05e-199))
		tmp = Float64(x_m / Float64(Float64(z - t) * z));
	else
		tmp = Float64(x_m / Float64(t * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -3.1e-108) || ~((z <= 2.05e-199)))
		tmp = x_m / ((z - t) * z);
	else
		tmp = x_m / (t * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -3.1e-108], N[Not[LessEqual[z, 2.05e-199]], $MachinePrecision]], N[(x$95$m / N[(N[(z - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.10000000000000014e-108 or 2.05000000000000011e-199 < z

   1. Initial program 89.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.0%

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

   if -3.10000000000000014e-108 < z < 2.05000000000000011e-199

   1. Initial program 96.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 78.2

         \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]

   5. Applied rewrites 78.2%

      \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-108} \lor \neg \left(z \leq 2.05 \cdot 10^{-199}\right):\\ \;\;\;\;\frac{x}{\left(z - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-139}:\\ \;\;\;\;\frac{x\_m}{\left(z - y\right) \cdot z}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-199}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(z - t\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -4e-139)
    (/ x_m (* (- z y) z))
    (if (<= z 2.05e-199) (/ x_m (* t y)) (/ x_m (* (- z t) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -4e-139) {
		tmp = x_m / ((z - y) * z);
	} else if (z <= 2.05e-199) {
		tmp = x_m / (t * y);
	} else {
		tmp = x_m / ((z - t) * z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-4d-139)) then
        tmp = x_m / ((z - y) * z)
    else if (z <= 2.05d-199) then
        tmp = x_m / (t * y)
    else
        tmp = x_m / ((z - t) * z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -4e-139) {
		tmp = x_m / ((z - y) * z);
	} else if (z <= 2.05e-199) {
		tmp = x_m / (t * y);
	} else {
		tmp = x_m / ((z - t) * z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -4e-139:
		tmp = x_m / ((z - y) * z)
	elif z <= 2.05e-199:
		tmp = x_m / (t * y)
	else:
		tmp = x_m / ((z - t) * z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -4e-139)
		tmp = Float64(x_m / Float64(Float64(z - y) * z));
	elseif (z <= 2.05e-199)
		tmp = Float64(x_m / Float64(t * y));
	else
		tmp = Float64(x_m / Float64(Float64(z - t) * z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -4e-139)
		tmp = x_m / ((z - y) * z);
	elseif (z <= 2.05e-199)
		tmp = x_m / (t * y);
	else
		tmp = x_m / ((z - t) * z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -4e-139], N[(x$95$m / N[(N[(z - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.05e-199], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(z - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -4.00000000000000012e-139

   1. Initial program 86.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.9%

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

   if -4.00000000000000012e-139 < z < 2.05000000000000011e-199

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 81.6

         \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]

   5. Applied rewrites 81.6%

      \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]

   if 2.05000000000000011e-199 < z

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-+r+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.1%

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

Alternative 9: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+125}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t 2.9e+125) (/ x_m (* (- y z) (- t z))) (/ (/ x_m (- y z)) t))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 2.9e+125) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.9d+125) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = (x_m / (y - z)) / t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 2.9e+125) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 2.9e+125:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = (x_m / (y - z)) / t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 2.9e+125)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / Float64(y - z)) / t);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 2.9e+125)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = (x_m / (y - z)) / t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 2.9e+125], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.89999999999999993e125

   1. Initial program 92.5%

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

   if 2.89999999999999993e125 < t

   1. Initial program 83.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 95.1

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

   5. Applied rewrites 95.1%

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

Alternative 10: 61.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9200000000000 \lor \neg \left(z \leq 1.12 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -9200000000000.0) (not (<= z 1.12e-11)))
    (/ x_m (* z z))
    (/ x_m (* t y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -9200000000000.0) || !(z <= 1.12e-11)) {
		tmp = x_m / (z * z);
	} else {
		tmp = x_m / (t * y);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-9200000000000.0d0)) .or. (.not. (z <= 1.12d-11))) then
        tmp = x_m / (z * z)
    else
        tmp = x_m / (t * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -9200000000000.0) || !(z <= 1.12e-11)) {
		tmp = x_m / (z * z);
	} else {
		tmp = x_m / (t * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -9200000000000.0) or not (z <= 1.12e-11):
		tmp = x_m / (z * z)
	else:
		tmp = x_m / (t * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -9200000000000.0) || !(z <= 1.12e-11))
		tmp = Float64(x_m / Float64(z * z));
	else
		tmp = Float64(x_m / Float64(t * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -9200000000000.0) || ~((z <= 1.12e-11)))
		tmp = x_m / (z * z);
	else
		tmp = x_m / (t * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -9200000000000.0], N[Not[LessEqual[z, 1.12e-11]], $MachinePrecision]], N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9.2e12 or 1.1200000000000001e-11 < z

   1. Initial program 85.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -9.2e12 < z < 1.1200000000000001e-11

   1. Initial program 96.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 57.1

         \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]

   5. Applied rewrites 57.1%

      \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9200000000000 \lor \neg \left(z \leq 1.12 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{x\_m}{y - z}}{t - z} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (/ (/ x_m (- y z)) (- t z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * ((x_m / (y - z)) / (t - z));
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * ((x_m / (y - z)) / (t - z))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * ((x_m / (y - z)) / (t - z));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * ((x_m / (y - z)) / (t - z))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(Float64(x_m / Float64(y - z)) / Float64(t - z)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * ((x_m / (y - z)) / (t - z));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 96.9

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

4. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{\frac{x}{y - z}}{t - z}} \]
5. Add Preprocessing

Alternative 12: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (/ x_m (* (- y z) (- t z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / ((y - z) * (t - z)));
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m / ((y - z) * (t - z)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / ((y - z) * (t - z)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / ((y - z) * (t - z)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / Float64(Float64(y - z) * Float64(t - z))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / ((y - z) * (t - z)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

Alternative 13: 38.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{t \cdot y} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m (* t y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (t * y));
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m / (t * y))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (t * y));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / (t * y))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / Float64(t * y)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / (t * y));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in z around 0

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

   1. lower-*.f64 37.2

      \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]

5. Applied rewrites 37.2%

   \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]
6. Add Preprocessing

Developer Target 1: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;\frac{x}{t\_1} < 0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))))
   (if (< (/ x t_1) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if ((x / t_1) < 0.0) {
		tmp = (x / (y - z)) / (t - z);
	} else {
		tmp = x * (1.0 / 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)
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) :: t_1
    real(8) :: tmp
    t_1 = (y - z) * (t - z)
    if ((x / t_1) < 0.0d0) then
        tmp = (x / (y - z)) / (t - z)
    else
        tmp = x * (1.0d0 / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if ((x / t_1) < 0.0) {
		tmp = (x / (y - z)) / (t - z);
	} else {
		tmp = x * (1.0 / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - z) * (t - z)
	tmp = 0
	if (x / t_1) < 0.0:
		tmp = (x / (y - z)) / (t - z)
	else:
		tmp = x * (1.0 / t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * Float64(t - z))
	tmp = 0.0
	if (Float64(x / t_1) < 0.0)
		tmp = Float64(Float64(x / Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(x * Float64(1.0 / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - z) * (t - z);
	tmp = 0.0;
	if ((x / t_1) < 0.0)
		tmp = (x / (y - z)) / (t - z);
	else
		tmp = x * (1.0 / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[Less[N[(x / t$95$1), $MachinePrecision], 0.0], N[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024352 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ x (* (- y z) (- t z))) 0) (/ (/ x (- y z)) (- t z)) (* x (/ 1 (* (- y z) (- t z))))))

  (/ x (* (- y z) (- t z))))