Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.0% → 97.8%

Time: 3.5s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

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(Float64(x * 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[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]

Initial Program: 84.0% accurate, 1.0× speedup

Math

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

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(Float64(x * 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[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y - z\right)}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t - z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{t - z} - \frac{z}{t - z}\right)\\ \end{array} \end{array} \end{array} \]

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 (- y z)) (- t z))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (* x_m (/ (- y z) (- t z)))
      (if (<= t_1 5e+138) t_1 (* x_m (- (/ y (- t z)) (/ 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 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x_m * ((y - z) / (t - z));
	} else if (t_1 <= 5e+138) {
		tmp = t_1;
	} else {
		tmp = x_m * ((y / (t - z)) - (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 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = x_m * ((y - z) / (t - z));
	} else if (t_1 <= 5e+138) {
		tmp = t_1;
	} else {
		tmp = x_m * ((y / (t - z)) - (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 = (x_m * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = x_m * ((y - z) / (t - z))
	elif t_1 <= 5e+138:
		tmp = t_1
	else:
		tmp = x_m * ((y / (t - z)) - (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(x_m * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x_m * Float64(Float64(y - z) / Float64(t - z)));
	elseif (t_1 <= 5e+138)
		tmp = t_1;
	else
		tmp = Float64(x_m * Float64(Float64(y / Float64(t - z)) - Float64(z / 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 = (x_m * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = x_m * ((y - z) / (t - z));
	elseif (t_1 <= 5e+138)
		tmp = t_1;
	else
		tmp = x_m * ((y / (t - z)) - (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[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+138], t$95$1, N[(x$95$m * N[(N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision] - N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 97.3

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

   3. Applied rewrites 97.3%

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

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 5.00000000000000016e138

   1. Initial program 84.0%

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

   if 5.00000000000000016e138 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 97.3

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

   3. Applied rewrites 97.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{y - z}{t - z}\\ t_2 := \frac{x\_m \cdot \left(y - z\right)}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ (- y z) (- t z)))) (t_2 (/ (* x_m (- y z)) (- t z))))
   (* x_s (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+138) t_2 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 * ((y - z) / (t - z));
	double t_2 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+138) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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 = x_m * ((y - z) / (t - z));
	double t_2 = (x_m * (y - z)) / (t - z);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+138) {
		tmp = t_2;
	} 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 * ((y - z) / (t - z))
	t_2 = (x_m * (y - z)) / (t - z)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+138:
		tmp = t_2
	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(Float64(y - z) / Float64(t - z)))
	t_2 = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+138)
		tmp = t_2;
	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 * ((y - z) / (t - z));
	t_2 = (x_m * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+138)
		tmp = t_2;
	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[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+138], t$95$2, t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -inf.0 or 5.00000000000000016e138 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 97.3

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

   3. Applied rewrites 97.3%

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

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 5.00000000000000016e138

   1. Initial program 84.0%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

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 84.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift--.f64 97.3

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

3. Applied rewrites 97.3%

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

Alternative 4: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-80}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{y \cdot x\_m}{t - z}\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+27}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* x_m (/ y z)))))
   (*
    x_s
    (if (<= z -1.14e+68)
      t_1
      (if (<= z -7e-80)
        (* x_m (/ y (- t z)))
        (if (<= z 1.8e-178)
          (/ (* y x_m) (- t z))
          (if (<= z 1.18e+27) (* x_m (/ (- y z) t)) 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 - (x_m * (y / z));
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= -7e-80) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 1.8e-178) {
		tmp = (y * x_m) / (t - z);
	} else if (z <= 1.18e+27) {
		tmp = x_m * ((y - z) / t);
	} 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 - (x_m * (y / z))
    if (z <= (-1.14d+68)) then
        tmp = t_1
    else if (z <= (-7d-80)) then
        tmp = x_m * (y / (t - z))
    else if (z <= 1.8d-178) then
        tmp = (y * x_m) / (t - z)
    else if (z <= 1.18d+27) then
        tmp = x_m * ((y - z) / t)
    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 - (x_m * (y / z));
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= -7e-80) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 1.8e-178) {
		tmp = (y * x_m) / (t - z);
	} else if (z <= 1.18e+27) {
		tmp = x_m * ((y - z) / t);
	} 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 - (x_m * (y / z))
	tmp = 0
	if z <= -1.14e+68:
		tmp = t_1
	elif z <= -7e-80:
		tmp = x_m * (y / (t - z))
	elif z <= 1.8e-178:
		tmp = (y * x_m) / (t - z)
	elif z <= 1.18e+27:
		tmp = x_m * ((y - z) / t)
	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(x_m * Float64(y / z)))
	tmp = 0.0
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= -7e-80)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	elseif (z <= 1.8e-178)
		tmp = Float64(Float64(y * x_m) / Float64(t - z));
	elseif (z <= 1.18e+27)
		tmp = Float64(x_m * Float64(Float64(y - z) / t));
	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 - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= -7e-80)
		tmp = x_m * (y / (t - z));
	elseif (z <= 1.8e-178)
		tmp = (y * x_m) / (t - z);
	elseif (z <= 1.18e+27)
		tmp = x_m * ((y - z) / t);
	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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.14e+68], t$95$1, If[LessEqual[z, -7e-80], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e-178], N[(N[(y * x$95$m), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.18e+27], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -1.13999999999999988e68 or 1.18000000000000006e27 < z

   1. Initial program 84.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -1.13999999999999988e68 < z < -7.00000000000000029e-80

   1. Initial program 84.0%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 52.7

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

   4. Applied rewrites 52.7%

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

   if -7.00000000000000029e-80 < z < 1.79999999999999997e-178

   1. Initial program 84.0%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.7

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

   4. Applied rewrites 49.7%

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

   if 1.79999999999999997e-178 < z < 1.18000000000000006e27

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 97.3

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 50.4%

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

Alternative 5: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-177}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+27}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* x_m (/ y z)))))
   (*
    x_s
    (if (<= z -1.14e+68)
      t_1
      (if (<= z -3.5e-177)
        (* x_m (/ y (- t z)))
        (if (<= z 1.18e+27) (/ (* x_m (- y z)) t) 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 - (x_m * (y / z));
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= -3.5e-177) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 1.18e+27) {
		tmp = (x_m * (y - z)) / t;
	} 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 - (x_m * (y / z))
    if (z <= (-1.14d+68)) then
        tmp = t_1
    else if (z <= (-3.5d-177)) then
        tmp = x_m * (y / (t - z))
    else if (z <= 1.18d+27) then
        tmp = (x_m * (y - z)) / t
    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 - (x_m * (y / z));
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= -3.5e-177) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 1.18e+27) {
		tmp = (x_m * (y - z)) / t;
	} 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 - (x_m * (y / z))
	tmp = 0
	if z <= -1.14e+68:
		tmp = t_1
	elif z <= -3.5e-177:
		tmp = x_m * (y / (t - z))
	elif z <= 1.18e+27:
		tmp = (x_m * (y - z)) / t
	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(x_m * Float64(y / z)))
	tmp = 0.0
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= -3.5e-177)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	elseif (z <= 1.18e+27)
		tmp = Float64(Float64(x_m * Float64(y - z)) / t);
	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 - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= -3.5e-177)
		tmp = x_m * (y / (t - z));
	elseif (z <= 1.18e+27)
		tmp = (x_m * (y - z)) / t;
	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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.14e+68], t$95$1, If[LessEqual[z, -3.5e-177], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.18e+27], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.13999999999999988e68 or 1.18000000000000006e27 < z

   1. Initial program 84.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -1.13999999999999988e68 < z < -3.5000000000000002e-177

   1. Initial program 84.0%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 52.7

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

   4. Applied rewrites 52.7%

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

   if -3.5000000000000002e-177 < z < 1.18000000000000006e27

   1. Initial program 84.0%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 47.1%

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

Alternative 6: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-178}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+27}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* x_m (/ y z)))))
   (*
    x_s
    (if (<= z -1.14e+68)
      t_1
      (if (<= z 8.5e-178)
        (* x_m (/ y (- t z)))
        (if (<= z 1.18e+27) (* x_m (/ (- y z) t)) 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 - (x_m * (y / z));
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= 8.5e-178) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 1.18e+27) {
		tmp = x_m * ((y - z) / t);
	} 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 - (x_m * (y / z))
    if (z <= (-1.14d+68)) then
        tmp = t_1
    else if (z <= 8.5d-178) then
        tmp = x_m * (y / (t - z))
    else if (z <= 1.18d+27) then
        tmp = x_m * ((y - z) / t)
    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 - (x_m * (y / z));
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= 8.5e-178) {
		tmp = x_m * (y / (t - z));
	} else if (z <= 1.18e+27) {
		tmp = x_m * ((y - z) / t);
	} 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 - (x_m * (y / z))
	tmp = 0
	if z <= -1.14e+68:
		tmp = t_1
	elif z <= 8.5e-178:
		tmp = x_m * (y / (t - z))
	elif z <= 1.18e+27:
		tmp = x_m * ((y - z) / t)
	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(x_m * Float64(y / z)))
	tmp = 0.0
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= 8.5e-178)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	elseif (z <= 1.18e+27)
		tmp = Float64(x_m * Float64(Float64(y - z) / t));
	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 - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= 8.5e-178)
		tmp = x_m * (y / (t - z));
	elseif (z <= 1.18e+27)
		tmp = x_m * ((y - z) / t);
	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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.14e+68], t$95$1, If[LessEqual[z, 8.5e-178], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.18e+27], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.13999999999999988e68 or 1.18000000000000006e27 < z

   1. Initial program 84.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -1.13999999999999988e68 < z < 8.5000000000000001e-178

   1. Initial program 84.0%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 52.7

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

   4. Applied rewrites 52.7%

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

   if 8.5000000000000001e-178 < z < 1.18000000000000006e27

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 97.3

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 50.4%

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

Alternative 7: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+27}:\\ \;\;\;\;x\_m \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* x_m (/ y z)))))
   (*
    x_s
    (if (<= z -1.14e+68) t_1 (if (<= z 1.8e+27) (* x_m (/ y (- t z))) 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 - (x_m * (y / z));
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= 1.8e+27) {
		tmp = x_m * (y / (t - z));
	} 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 - (x_m * (y / z))
    if (z <= (-1.14d+68)) then
        tmp = t_1
    else if (z <= 1.8d+27) then
        tmp = x_m * (y / (t - z))
    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 - (x_m * (y / z));
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= 1.8e+27) {
		tmp = x_m * (y / (t - z));
	} 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 - (x_m * (y / z))
	tmp = 0
	if z <= -1.14e+68:
		tmp = t_1
	elif z <= 1.8e+27:
		tmp = x_m * (y / (t - z))
	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(x_m * Float64(y / z)))
	tmp = 0.0
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= 1.8e+27)
		tmp = Float64(x_m * Float64(y / Float64(t - z)));
	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 - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= 1.8e+27)
		tmp = x_m * (y / (t - z));
	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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.14e+68], t$95$1, If[LessEqual[z, 1.8e+27], N[(x$95$m * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.13999999999999988e68 or 1.79999999999999991e27 < z

   1. Initial program 84.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -1.13999999999999988e68 < z < 1.79999999999999991e27

   1. Initial program 84.0%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 52.7

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

   4. Applied rewrites 52.7%

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

Alternative 8: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m - x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* x_m (/ y z)))))
   (* x_s (if (<= z -6.5e-66) t_1 (if (<= z 2.3e-12) (* y (/ x_m t)) 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 - (x_m * (y / z));
	double tmp;
	if (z <= -6.5e-66) {
		tmp = t_1;
	} else if (z <= 2.3e-12) {
		tmp = y * (x_m / t);
	} 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 - (x_m * (y / z))
    if (z <= (-6.5d-66)) then
        tmp = t_1
    else if (z <= 2.3d-12) then
        tmp = y * (x_m / t)
    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 - (x_m * (y / z));
	double tmp;
	if (z <= -6.5e-66) {
		tmp = t_1;
	} else if (z <= 2.3e-12) {
		tmp = y * (x_m / t);
	} 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 - (x_m * (y / z))
	tmp = 0
	if z <= -6.5e-66:
		tmp = t_1
	elif z <= 2.3e-12:
		tmp = y * (x_m / t)
	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(x_m * Float64(y / z)))
	tmp = 0.0
	if (z <= -6.5e-66)
		tmp = t_1;
	elseif (z <= 2.3e-12)
		tmp = Float64(y * Float64(x_m / t));
	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 - (x_m * (y / z));
	tmp = 0.0;
	if (z <= -6.5e-66)
		tmp = t_1;
	elseif (z <= 2.3e-12)
		tmp = y * (x_m / t);
	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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -6.5e-66], t$95$1, If[LessEqual[z, 2.3e-12], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000024e-66 or 2.29999999999999989e-12 < z

   1. Initial program 84.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -6.50000000000000024e-66 < z < 2.29999999999999989e-12

   1. Initial program 84.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.3

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

   4. Applied rewrites 37.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

Alternative 9: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(z - y\right)}{z}\\ \end{array} \end{array} \]

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 -3.1e+75)
    (fma t (/ x_m z) x_m)
    (if (<= z 2.3e-12) (* y (/ x_m t)) (/ (* x_m (- z y)) 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 <= -3.1e+75) {
		tmp = fma(t, (x_m / z), x_m);
	} else if (z <= 2.3e-12) {
		tmp = y * (x_m / t);
	} else {
		tmp = (x_m * (z - y)) / 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 <= -3.1e+75)
		tmp = fma(t, Float64(x_m / z), x_m);
	elseif (z <= 2.3e-12)
		tmp = Float64(y * Float64(x_m / t));
	else
		tmp = Float64(Float64(x_m * Float64(z - y)) / 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_] := N[(x$95$s * If[LessEqual[z, -3.1e+75], N[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision], If[LessEqual[z, 2.3e-12], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(z - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.1000000000000001e75

   1. Initial program 84.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if -3.1000000000000001e75 < z < 2.29999999999999989e-12

   1. Initial program 84.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.3

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

   4. Applied rewrites 37.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

   if 2.29999999999999989e-12 < z

   1. Initial program 84.0%

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

   2. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-out-- N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 45.3

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

   7. Applied rewrites 45.3%

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

Alternative 10: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{x\_m}{z}, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (fma t (/ x_m z) x_m)))
   (* x_s (if (<= z -3.1e+75) t_1 (if (<= z 3.8e+24) (* y (/ x_m t)) 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 = fma(t, (x_m / z), x_m);
	double tmp;
	if (z <= -3.1e+75) {
		tmp = t_1;
	} else if (z <= 3.8e+24) {
		tmp = y * (x_m / t);
	} 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 = fma(t, Float64(x_m / z), x_m)
	tmp = 0.0
	if (z <= -3.1e+75)
		tmp = t_1;
	elseif (z <= 3.8e+24)
		tmp = Float64(y * Float64(x_m / t));
	else
		tmp = t_1;
	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[(t * N[(x$95$m / z), $MachinePrecision] + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.1e+75], t$95$1, If[LessEqual[z, 3.8e+24], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1000000000000001e75 or 3.80000000000000015e24 < z

   1. Initial program 84.0%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if -3.1000000000000001e75 < z < 3.80000000000000015e24

   1. Initial program 84.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.3

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

   4. Applied rewrites 37.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

Alternative 11: 59.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := -\left(-x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.14 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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))))
   (* x_s (if (<= z -1.14e+68) t_1 (if (<= z 9e-9) (* y (/ x_m t)) 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);
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= 9e-9) {
		tmp = y * (x_m / t);
	} 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)
    if (z <= (-1.14d+68)) then
        tmp = t_1
    else if (z <= 9d-9) then
        tmp = y * (x_m / t)
    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);
	double tmp;
	if (z <= -1.14e+68) {
		tmp = t_1;
	} else if (z <= 9e-9) {
		tmp = y * (x_m / t);
	} 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)
	tmp = 0
	if z <= -1.14e+68:
		tmp = t_1
	elif z <= 9e-9:
		tmp = y * (x_m / t)
	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(-Float64(-x_m))
	tmp = 0.0
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= 9e-9)
		tmp = Float64(y * Float64(x_m / t));
	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);
	tmp = 0.0;
	if (z <= -1.14e+68)
		tmp = t_1;
	elseif (z <= 9e-9)
		tmp = y * (x_m / t);
	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 = (-(-x$95$m))}, N[(x$95$s * If[LessEqual[z, -1.14e+68], t$95$1, If[LessEqual[z, 9e-9], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.13999999999999988e68 or 8.99999999999999953e-9 < z

   1. Initial program 84.0%

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

   2. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 35.5

         \[\leadsto -\left(-x\right) \]

   7. Applied rewrites 35.5%

      \[\leadsto -\left(-x\right) \]

   if -1.13999999999999988e68 < z < 8.99999999999999953e-9

   1. Initial program 84.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.3

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

   4. Applied rewrites 37.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

Alternative 12: 35.5% accurate, 4.3× speedup

Math

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

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))))

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);
}

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)
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);
}

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)

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)))
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);
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 * (-(-x$95$m))), $MachinePrecision]

Derivation

1. Initial program 84.0%

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

2. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 45.3

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

4. Applied rewrites 45.3%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.5

      \[\leadsto -\left(-x\right) \]

7. Applied rewrites 35.5%

   \[\leadsto -\left(-x\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64
  (/ (* x (- y z)) (- t z)))