Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Percentage Accurate: 89.6% → 98.4%

Time: 4.3s

Alternatives: 18

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

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) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

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) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + y \cdot \frac{z}{t\_1}}{x + 1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -400000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.9999999697559053:\\ \;\;\;\;\frac{\left(-\frac{\left(-y\right) - \frac{-x}{z}}{t}\right) + x}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (+ x (* y (/ z t_1))) (+ x 1.0)))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_3 -400000000000.0)
     t_2
     (if (<= t_3 0.9999999697559053)
       (/ (+ (- (/ (- (- y) (/ (- x) z)) t)) x) (+ x 1.0))
       (if (<= t_3 2.0)
         1.0
         (if (<= t_3 INFINITY) t_2 (/ (+ x (/ y t)) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x + 1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -400000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.9999999697559053) {
		tmp = (-((-y - (-x / z)) / t) + x) / (x + 1.0);
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x + 1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -400000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.9999999697559053) {
		tmp = (-((-y - (-x / z)) / t) + x) / (x + 1.0);
	} else if (t_3 <= 2.0) {
		tmp = 1.0;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (y * (z / t_1))) / (x + 1.0)
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_3 <= -400000000000.0:
		tmp = t_2
	elif t_3 <= 0.9999999697559053:
		tmp = (-((-y - (-x / z)) / t) + x) / (x + 1.0)
	elif t_3 <= 2.0:
		tmp = 1.0
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(y * Float64(z / t_1))) / Float64(x + 1.0))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -400000000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.9999999697559053)
		tmp = Float64(Float64(Float64(-Float64(Float64(Float64(-y) - Float64(Float64(-x) / z)) / t)) + x) / Float64(x + 1.0));
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (y * (z / t_1))) / (x + 1.0);
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -400000000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.9999999697559053)
		tmp = (-((-y - (-x / z)) / t) + x) / (x + 1.0);
	elseif (t_3 <= 2.0)
		tmp = 1.0;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -400000000000.0], t$95$2, If[LessEqual[t$95$3, 0.9999999697559053], N[(N[((-N[(N[((-y) - N[((-x) / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], 1.0, If[LessEqual[t$95$3, Infinity], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4e11 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 78.4%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if -4e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999999969755905327

   1. Initial program 96.1%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if 0.999999969755905327 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{1} \]

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + y \cdot \frac{z}{t\_1}}{x + 1}\\ t_3 := x + \frac{y \cdot z - x}{t\_1}\\ t_4 := \frac{t\_3}{x + 1}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_3}{1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (+ x (* y (/ z t_1))) (+ x 1.0)))
        (t_3 (+ x (/ (- (* y z) x) t_1)))
        (t_4 (/ t_3 (+ x 1.0))))
   (if (<= t_4 -1e+14)
     t_2
     (if (<= t_4 5e-17)
       (/ t_3 1.0)
       (if (<= t_4 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_2 (/ (+ x (/ y t)) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x + 1.0);
	double t_3 = x + (((y * z) - x) / t_1);
	double t_4 = t_3 / (x + 1.0);
	double tmp;
	if (t_4 <= -1e+14) {
		tmp = t_2;
	} else if (t_4 <= 5e-17) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x + 1.0);
	double t_3 = x + (((y * z) - x) / t_1);
	double t_4 = t_3 / (x + 1.0);
	double tmp;
	if (t_4 <= -1e+14) {
		tmp = t_2;
	} else if (t_4 <= 5e-17) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (y * (z / t_1))) / (x + 1.0)
	t_3 = x + (((y * z) - x) / t_1)
	t_4 = t_3 / (x + 1.0)
	tmp = 0
	if t_4 <= -1e+14:
		tmp = t_2
	elif t_4 <= 5e-17:
		tmp = t_3 / 1.0
	elif t_4 <= 2.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_4 <= math.inf:
		tmp = t_2
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(y * Float64(z / t_1))) / Float64(x + 1.0))
	t_3 = Float64(x + Float64(Float64(Float64(y * z) - x) / t_1))
	t_4 = Float64(t_3 / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -1e+14)
		tmp = t_2;
	elseif (t_4 <= 5e-17)
		tmp = Float64(t_3 / 1.0);
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (y * (z / t_1))) / (x + 1.0);
	t_3 = x + (((y * z) - x) / t_1);
	t_4 = t_3 / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -1e+14)
		tmp = t_2;
	elseif (t_4 <= 5e-17)
		tmp = t_3 / 1.0;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e+14], t$95$2, If[LessEqual[t$95$4, 5e-17], N[(t$95$3 / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e14 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 78.4%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if -1e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e-17

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 94.7%

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

   if 4.9999999999999999e-17 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \left(1 + x\right) \cdot t\_1\\ \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t\_2}, \frac{x}{1 + x}\right) - \frac{x}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (* (+ 1.0 x) t_1)))
   (if (<= (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0)) INFINITY)
     (- (fma y (/ z t_2) (/ x (+ 1.0 x))) (/ x t_2))
     (/ (+ x (/ y t)) (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (1.0 + x) * t_1;
	double tmp;
	if (((x + (((y * z) - x) / t_1)) / (x + 1.0)) <= ((double) INFINITY)) {
		tmp = fma(y, (z / t_2), (x / (1.0 + x))) - (x / t_2);
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(1.0 + x) * t_1)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0)) <= Inf)
		tmp = Float64(fma(y, Float64(z / t_2), Float64(x / Float64(1.0 + x))) - Float64(x / t_2));
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y * N[(z / t$95$2), $MachinePrecision] + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 93.4%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 98.0%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 4: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{t\_1}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{x + \frac{t\_1}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ t_1 t_2)) (+ x 1.0))))
   (if (<= t_3 -1e+14)
     (/ (+ x (* y (/ z t_2))) (+ x 1.0))
     (if (<= t_3 4e+291)
       (/ (+ x (/ t_1 (fma t z (- x)))) (+ x 1.0))
       (+ (/ x (+ 1.0 x)) (/ y (* t (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (t * z) - x;
	double t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1e+14) {
		tmp = (x + (y * (z / t_2))) / (x + 1.0);
	} else if (t_3 <= 4e+291) {
		tmp = (x + (t_1 / fma(t, z, -x))) / (x + 1.0);
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - x)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -1e+14)
		tmp = Float64(Float64(x + Float64(y * Float64(z / t_2))) / Float64(x + 1.0));
	elseif (t_3 <= 4e+291)
		tmp = Float64(Float64(x + Float64(t_1 / fma(t, z, Float64(-x)))) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(y / Float64(t * Float64(1.0 + x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+14], N[(N[(x + N[(y * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+291], N[(N[(x + N[(t$95$1 / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e14

   1. Initial program 77.4%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if -1e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 84.6

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

   7. Applied rewrites 84.6%

      \[\leadsto \frac{x}{1 + x} + \color{blue}{\frac{y}{t \cdot \left(1 + x\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -1e+14)
     (/ (+ x (* y (/ z t_1))) (+ x 1.0))
     (if (<= t_2 4e+291) t_2 (+ (/ x (+ 1.0 x)) (/ y (* t (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -1e+14) {
		tmp = (x + (y * (z / t_1))) / (x + 1.0);
	} else if (t_2 <= 4e+291) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_2 <= (-1d+14)) then
        tmp = (x + (y * (z / t_1))) / (x + 1.0d0)
    else if (t_2 <= 4d+291) then
        tmp = t_2
    else
        tmp = (x / (1.0d0 + x)) + (y / (t * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -1e+14) {
		tmp = (x + (y * (z / t_1))) / (x + 1.0);
	} else if (t_2 <= 4e+291) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -1e+14:
		tmp = (x + (y * (z / t_1))) / (x + 1.0)
	elif t_2 <= 4e+291:
		tmp = t_2
	else:
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1e+14)
		tmp = Float64(Float64(x + Float64(y * Float64(z / t_1))) / Float64(x + 1.0));
	elseif (t_2 <= 4e+291)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(y / Float64(t * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -1e+14)
		tmp = (x + (y * (z / t_1))) / (x + 1.0);
	elseif (t_2 <= 4e+291)
		tmp = t_2;
	else
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+14], N[(N[(x + N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+291], t$95$2, N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e14

   1. Initial program 77.4%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if -1e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 98.9%

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

   if 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 84.6

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

   7. Applied rewrites 84.6%

      \[\leadsto \frac{x}{1 + x} + \color{blue}{\frac{y}{t \cdot \left(1 + x\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{y \cdot \frac{z}{t\_1}}{x + 1}\\ t_3 := x + \frac{y \cdot z - x}{t\_1}\\ t_4 := \frac{t\_3}{x + 1}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_3}{1}\\ \mathbf{elif}\;t\_4 \leq 4:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (* y (/ z t_1)) (+ x 1.0)))
        (t_3 (+ x (/ (- (* y z) x) t_1)))
        (t_4 (/ t_3 (+ x 1.0))))
   (if (<= t_4 -1e+14)
     t_2
     (if (<= t_4 5e-17)
       (/ t_3 1.0)
       (if (<= t_4 4.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_4 4e+291)
           t_2
           (+ (/ x (+ 1.0 x)) (/ y (* t (+ 1.0 x))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (y * (z / t_1)) / (x + 1.0);
	double t_3 = x + (((y * z) - x) / t_1);
	double t_4 = t_3 / (x + 1.0);
	double tmp;
	if (t_4 <= -1e+14) {
		tmp = t_2;
	} else if (t_4 <= 5e-17) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 4.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_4 <= 4e+291) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = (y * (z / t_1)) / (x + 1.0d0)
    t_3 = x + (((y * z) - x) / t_1)
    t_4 = t_3 / (x + 1.0d0)
    if (t_4 <= (-1d+14)) then
        tmp = t_2
    else if (t_4 <= 5d-17) then
        tmp = t_3 / 1.0d0
    else if (t_4 <= 4.0d0) then
        tmp = (x - (x / t_1)) / (x + 1.0d0)
    else if (t_4 <= 4d+291) then
        tmp = t_2
    else
        tmp = (x / (1.0d0 + x)) + (y / (t * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (y * (z / t_1)) / (x + 1.0);
	double t_3 = x + (((y * z) - x) / t_1);
	double t_4 = t_3 / (x + 1.0);
	double tmp;
	if (t_4 <= -1e+14) {
		tmp = t_2;
	} else if (t_4 <= 5e-17) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 4.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_4 <= 4e+291) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (y * (z / t_1)) / (x + 1.0)
	t_3 = x + (((y * z) - x) / t_1)
	t_4 = t_3 / (x + 1.0)
	tmp = 0
	if t_4 <= -1e+14:
		tmp = t_2
	elif t_4 <= 5e-17:
		tmp = t_3 / 1.0
	elif t_4 <= 4.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_4 <= 4e+291:
		tmp = t_2
	else:
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(y * Float64(z / t_1)) / Float64(x + 1.0))
	t_3 = Float64(x + Float64(Float64(Float64(y * z) - x) / t_1))
	t_4 = Float64(t_3 / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -1e+14)
		tmp = t_2;
	elseif (t_4 <= 5e-17)
		tmp = Float64(t_3 / 1.0);
	elseif (t_4 <= 4.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_4 <= 4e+291)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(y / Float64(t * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (y * (z / t_1)) / (x + 1.0);
	t_3 = x + (((y * z) - x) / t_1);
	t_4 = t_3 / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -1e+14)
		tmp = t_2;
	elseif (t_4 <= 5e-17)
		tmp = t_3 / 1.0;
	elseif (t_4 <= 4.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_4 <= 4e+291)
		tmp = t_2;
	else
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e+14], t$95$2, If[LessEqual[t$95$4, 5e-17], N[(t$95$3 / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 4.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+291], t$95$2, N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1e14 or 4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 86.3%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

   if -1e14 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e-17

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 94.7%

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

   if 4.9999999999999999e-17 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 84.6

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

   7. Applied rewrites 84.6%

      \[\leadsto \frac{x}{1 + x} + \color{blue}{\frac{y}{t \cdot \left(1 + x\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{y \cdot \frac{z}{t\_1}}{x + 1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_3 \leq 4:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (* y (/ z t_1)) (+ x 1.0)))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_3 -5e-5)
     t_2
     (if (<= t_3 5e-17)
       (/ (+ x (/ y t)) 1.0)
       (if (<= t_3 4.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_3 4e+291)
           t_2
           (+ (/ x (+ 1.0 x)) (/ y (* t (+ 1.0 x))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (y * (z / t_1)) / (x + 1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e-5) {
		tmp = t_2;
	} else if (t_3 <= 5e-17) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_3 <= 4.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= 4e+291) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = (y * (z / t_1)) / (x + 1.0d0)
    t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_3 <= (-5d-5)) then
        tmp = t_2
    else if (t_3 <= 5d-17) then
        tmp = (x + (y / t)) / 1.0d0
    else if (t_3 <= 4.0d0) then
        tmp = (x - (x / t_1)) / (x + 1.0d0)
    else if (t_3 <= 4d+291) then
        tmp = t_2
    else
        tmp = (x / (1.0d0 + x)) + (y / (t * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (y * (z / t_1)) / (x + 1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e-5) {
		tmp = t_2;
	} else if (t_3 <= 5e-17) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_3 <= 4.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= 4e+291) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (y * (z / t_1)) / (x + 1.0)
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_3 <= -5e-5:
		tmp = t_2
	elif t_3 <= 5e-17:
		tmp = (x + (y / t)) / 1.0
	elif t_3 <= 4.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_3 <= 4e+291:
		tmp = t_2
	else:
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(y * Float64(z / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -5e-5)
		tmp = t_2;
	elseif (t_3 <= 5e-17)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_3 <= 4.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_3 <= 4e+291)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(y / Float64(t * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (y * (z / t_1)) / (x + 1.0);
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -5e-5)
		tmp = t_2;
	elseif (t_3 <= 5e-17)
		tmp = (x + (y / t)) / 1.0;
	elseif (t_3 <= 4.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_3 <= 4e+291)
		tmp = t_2;
	else
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-5], t$95$2, If[LessEqual[t$95$3, 5e-17], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 4.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+291], t$95$2, N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000024e-5 or 4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 86.7%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

   if -5.00000000000000024e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e-17

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 85.8

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.4%

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

   if 4.9999999999999999e-17 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 84.6

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

   7. Applied rewrites 84.6%

      \[\leadsto \frac{x}{1 + x} + \color{blue}{\frac{y}{t \cdot \left(1 + x\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+79}:\\ \;\;\;\;\frac{z}{1 + x} \cdot \frac{y}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_2 \leq 4:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{z \cdot y}{\left(1 + x\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -2e+79)
     (* (/ z (+ 1.0 x)) (/ y t_1))
     (if (<= t_2 5e-17)
       (/ (+ x (/ y t)) 1.0)
       (if (<= t_2 4.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_2 4e+291)
           (/ (* z y) (* (+ 1.0 x) t_1))
           (+ (/ x (+ 1.0 x)) (/ y (* t (+ 1.0 x))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+79) {
		tmp = (z / (1.0 + x)) * (y / t_1);
	} else if (t_2 <= 5e-17) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_2 <= 4.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_2 <= 4e+291) {
		tmp = (z * y) / ((1.0 + x) * t_1);
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_2 <= (-2d+79)) then
        tmp = (z / (1.0d0 + x)) * (y / t_1)
    else if (t_2 <= 5d-17) then
        tmp = (x + (y / t)) / 1.0d0
    else if (t_2 <= 4.0d0) then
        tmp = (x - (x / t_1)) / (x + 1.0d0)
    else if (t_2 <= 4d+291) then
        tmp = (z * y) / ((1.0d0 + x) * t_1)
    else
        tmp = (x / (1.0d0 + x)) + (y / (t * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e+79) {
		tmp = (z / (1.0 + x)) * (y / t_1);
	} else if (t_2 <= 5e-17) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_2 <= 4.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_2 <= 4e+291) {
		tmp = (z * y) / ((1.0 + x) * t_1);
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -2e+79:
		tmp = (z / (1.0 + x)) * (y / t_1)
	elif t_2 <= 5e-17:
		tmp = (x + (y / t)) / 1.0
	elif t_2 <= 4.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_2 <= 4e+291:
		tmp = (z * y) / ((1.0 + x) * t_1)
	else:
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e+79)
		tmp = Float64(Float64(z / Float64(1.0 + x)) * Float64(y / t_1));
	elseif (t_2 <= 5e-17)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_2 <= 4.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_2 <= 4e+291)
		tmp = Float64(Float64(z * y) / Float64(Float64(1.0 + x) * t_1));
	else
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(y / Float64(t * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -2e+79)
		tmp = (z / (1.0 + x)) * (y / t_1);
	elseif (t_2 <= 5e-17)
		tmp = (x + (y / t)) / 1.0;
	elseif (t_2 <= 4.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_2 <= 4e+291)
		tmp = (z * y) / ((1.0 + x) * t_1);
	else
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+79], N[(N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-17], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 4.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+291], N[(N[(z * y), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.99999999999999993e79

   1. Initial program 72.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 72.0

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

   4. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 80.6

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

   6. Applied rewrites 80.6%

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

   if -1.99999999999999993e79 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e-17

   1. Initial program 96.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 82.9

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 81.6%

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

   if 4.9999999999999999e-17 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if 4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 84.6

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

   7. Applied rewrites 84.6%

      \[\leadsto \frac{x}{1 + x} + \color{blue}{\frac{y}{t \cdot \left(1 + x\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{z \cdot y}{\left(1 + x\right) \cdot t\_1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_3 \leq 4:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (* z y) (* (+ 1.0 x) t_1)))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_3 -5e-5)
     t_2
     (if (<= t_3 5e-17)
       (/ (+ x (/ y t)) 1.0)
       (if (<= t_3 4.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_3 4e+291)
           t_2
           (+ (/ x (+ 1.0 x)) (/ y (* t (+ 1.0 x))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (z * y) / ((1.0 + x) * t_1);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e-5) {
		tmp = t_2;
	} else if (t_3 <= 5e-17) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_3 <= 4.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= 4e+291) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = (z * y) / ((1.0d0 + x) * t_1)
    t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_3 <= (-5d-5)) then
        tmp = t_2
    else if (t_3 <= 5d-17) then
        tmp = (x + (y / t)) / 1.0d0
    else if (t_3 <= 4.0d0) then
        tmp = (x - (x / t_1)) / (x + 1.0d0)
    else if (t_3 <= 4d+291) then
        tmp = t_2
    else
        tmp = (x / (1.0d0 + x)) + (y / (t * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (z * y) / ((1.0 + x) * t_1);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e-5) {
		tmp = t_2;
	} else if (t_3 <= 5e-17) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_3 <= 4.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_3 <= 4e+291) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (z * y) / ((1.0 + x) * t_1)
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_3 <= -5e-5:
		tmp = t_2
	elif t_3 <= 5e-17:
		tmp = (x + (y / t)) / 1.0
	elif t_3 <= 4.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_3 <= 4e+291:
		tmp = t_2
	else:
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(z * y) / Float64(Float64(1.0 + x) * t_1))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -5e-5)
		tmp = t_2;
	elseif (t_3 <= 5e-17)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_3 <= 4.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_3 <= 4e+291)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(y / Float64(t * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (z * y) / ((1.0 + x) * t_1);
	t_3 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -5e-5)
		tmp = t_2;
	elseif (t_3 <= 5e-17)
		tmp = (x + (y / t)) / 1.0;
	elseif (t_3 <= 4.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_3 <= 4e+291)
		tmp = t_2;
	else
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-5], t$95$2, If[LessEqual[t$95$3, 5e-17], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 4.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+291], t$95$2, N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000024e-5 or 4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 86.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 84.8

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

   4. Applied rewrites 84.8%

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

   if -5.00000000000000024e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e-17

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 85.8

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.4%

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

   if 4.9999999999999999e-17 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 84.6

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

   7. Applied rewrites 84.6%

      \[\leadsto \frac{x}{1 + x} + \color{blue}{\frac{y}{t \cdot \left(1 + x\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{t}\\ t_2 := t \cdot z - x\\ t_3 := \frac{z \cdot y}{\left(1 + x\right) \cdot t\_2}\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{elif}\;t\_4 \leq 4:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y t)))
        (t_2 (- (* t z) x))
        (t_3 (/ (* z y) (* (+ 1.0 x) t_2)))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -5e-5)
     t_3
     (if (<= t_4 5e-17)
       (/ t_1 1.0)
       (if (<= t_4 4.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_4 4e+291) t_3 (/ t_1 (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / t);
	double t_2 = (t * z) - x;
	double t_3 = (z * y) / ((1.0 + x) * t_2);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -5e-5) {
		tmp = t_3;
	} else if (t_4 <= 5e-17) {
		tmp = t_1 / 1.0;
	} else if (t_4 <= 4.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_4 <= 4e+291) {
		tmp = t_3;
	} else {
		tmp = t_1 / (x + 1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = x + (y / t)
    t_2 = (t * z) - x
    t_3 = (z * y) / ((1.0d0 + x) * t_2)
    t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    if (t_4 <= (-5d-5)) then
        tmp = t_3
    else if (t_4 <= 5d-17) then
        tmp = t_1 / 1.0d0
    else if (t_4 <= 4.0d0) then
        tmp = (x - (x / t_2)) / (x + 1.0d0)
    else if (t_4 <= 4d+291) then
        tmp = t_3
    else
        tmp = t_1 / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (y / t);
	double t_2 = (t * z) - x;
	double t_3 = (z * y) / ((1.0 + x) * t_2);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -5e-5) {
		tmp = t_3;
	} else if (t_4 <= 5e-17) {
		tmp = t_1 / 1.0;
	} else if (t_4 <= 4.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_4 <= 4e+291) {
		tmp = t_3;
	} else {
		tmp = t_1 / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (y / t)
	t_2 = (t * z) - x
	t_3 = (z * y) / ((1.0 + x) * t_2)
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -5e-5:
		tmp = t_3
	elif t_4 <= 5e-17:
		tmp = t_1 / 1.0
	elif t_4 <= 4.0:
		tmp = (x - (x / t_2)) / (x + 1.0)
	elif t_4 <= 4e+291:
		tmp = t_3
	else:
		tmp = t_1 / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / t))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(z * y) / Float64(Float64(1.0 + x) * t_2))
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -5e-5)
		tmp = t_3;
	elseif (t_4 <= 5e-17)
		tmp = Float64(t_1 / 1.0);
	elseif (t_4 <= 4.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_4 <= 4e+291)
		tmp = t_3;
	else
		tmp = Float64(t_1 / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / t);
	t_2 = (t * z) - x;
	t_3 = (z * y) / ((1.0 + x) * t_2);
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -5e-5)
		tmp = t_3;
	elseif (t_4 <= 5e-17)
		tmp = t_1 / 1.0;
	elseif (t_4 <= 4.0)
		tmp = (x - (x / t_2)) / (x + 1.0);
	elseif (t_4 <= 4e+291)
		tmp = t_3;
	else
		tmp = t_1 / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e-5], t$95$3, If[LessEqual[t$95$4, 5e-17], N[(t$95$1 / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 4.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+291], t$95$3, N[(t$95$1 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000024e-5 or 4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 86.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 84.8

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

   4. Applied rewrites 84.8%

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

   if -5.00000000000000024e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999999e-17

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 85.8

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.4%

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

   if 4.9999999999999999e-17 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 26.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 83.9

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

   4. Applied rewrites 83.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{z \cdot y}{\left(1 + x\right) \cdot t\_2}\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 0.9999999697559053:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 4:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* t z) x))
        (t_3 (/ (* z y) (* (+ 1.0 x) t_2)))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -5e-5)
     t_3
     (if (<= t_4 0.9999999697559053)
       t_1
       (if (<= t_4 4.0) 1.0 (if (<= t_4 4e+291) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (z * y) / ((1.0 + x) * t_2);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -5e-5) {
		tmp = t_3;
	} else if (t_4 <= 0.9999999697559053) {
		tmp = t_1;
	} else if (t_4 <= 4.0) {
		tmp = 1.0;
	} else if (t_4 <= 4e+291) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (t * z) - x
    t_3 = (z * y) / ((1.0d0 + x) * t_2)
    t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    if (t_4 <= (-5d-5)) then
        tmp = t_3
    else if (t_4 <= 0.9999999697559053d0) then
        tmp = t_1
    else if (t_4 <= 4.0d0) then
        tmp = 1.0d0
    else if (t_4 <= 4d+291) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (z * y) / ((1.0 + x) * t_2);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -5e-5) {
		tmp = t_3;
	} else if (t_4 <= 0.9999999697559053) {
		tmp = t_1;
	} else if (t_4 <= 4.0) {
		tmp = 1.0;
	} else if (t_4 <= 4e+291) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (t * z) - x
	t_3 = (z * y) / ((1.0 + x) * t_2)
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -5e-5:
		tmp = t_3
	elif t_4 <= 0.9999999697559053:
		tmp = t_1
	elif t_4 <= 4.0:
		tmp = 1.0
	elif t_4 <= 4e+291:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(z * y) / Float64(Float64(1.0 + x) * t_2))
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -5e-5)
		tmp = t_3;
	elseif (t_4 <= 0.9999999697559053)
		tmp = t_1;
	elseif (t_4 <= 4.0)
		tmp = 1.0;
	elseif (t_4 <= 4e+291)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (t * z) - x;
	t_3 = (z * y) / ((1.0 + x) * t_2);
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -5e-5)
		tmp = t_3;
	elseif (t_4 <= 0.9999999697559053)
		tmp = t_1;
	elseif (t_4 <= 4.0)
		tmp = 1.0;
	elseif (t_4 <= 4e+291)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * y), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e-5], t$95$3, If[LessEqual[t$95$4, 0.9999999697559053], t$95$1, If[LessEqual[t$95$4, 4.0], 1.0, If[LessEqual[t$95$4, 4e+291], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000024e-5 or 4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 86.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 84.8

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

   4. Applied rewrites 84.8%

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

   if -5.00000000000000024e-5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999999969755905327 or 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 84.7

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

   4. Applied rewrites 84.7%

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

   if 0.999999969755905327 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq 0.9999999697559053:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{z \cdot y}{1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 0.9999999697559053)
     t_1
     (if (<= t_3 5e+21)
       1.0
       (if (<= t_3 4e+291) (/ (* z y) (* 1.0 t_2)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= 0.9999999697559053) {
		tmp = t_1;
	} else if (t_3 <= 5e+21) {
		tmp = 1.0;
	} else if (t_3 <= 4e+291) {
		tmp = (z * y) / (1.0 * t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (t * z) - x
    t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    if (t_3 <= 0.9999999697559053d0) then
        tmp = t_1
    else if (t_3 <= 5d+21) then
        tmp = 1.0d0
    else if (t_3 <= 4d+291) then
        tmp = (z * y) / (1.0d0 * t_2)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= 0.9999999697559053) {
		tmp = t_1;
	} else if (t_3 <= 5e+21) {
		tmp = 1.0;
	} else if (t_3 <= 4e+291) {
		tmp = (z * y) / (1.0 * t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= 0.9999999697559053:
		tmp = t_1
	elif t_3 <= 5e+21:
		tmp = 1.0
	elif t_3 <= 4e+291:
		tmp = (z * y) / (1.0 * t_2)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= 0.9999999697559053)
		tmp = t_1;
	elseif (t_3 <= 5e+21)
		tmp = 1.0;
	elseif (t_3 <= 4e+291)
		tmp = Float64(Float64(z * y) / Float64(1.0 * t_2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= 0.9999999697559053)
		tmp = t_1;
	elseif (t_3 <= 5e+21)
		tmp = 1.0;
	elseif (t_3 <= 4e+291)
		tmp = (z * y) / (1.0 * t_2);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.9999999697559053], t$95$1, If[LessEqual[t$95$3, 5e+21], 1.0, If[LessEqual[t$95$3, 4e+291], N[(N[(z * y), $MachinePrecision] / N[(1.0 * t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999999969755905327 or 3.9999999999999998e291 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 78.2

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

   4. Applied rewrites 78.2%

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

   if 0.999999969755905327 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e21

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 97.5%

      \[\leadsto \color{blue}{1} \]

   if 5e21 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e291

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.9%

      \[\leadsto \frac{z \cdot y}{1 \cdot \left(\color{blue}{t \cdot z} - x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 0.9999999697559053:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 0.9999999697559053) t_1 (if (<= t_2 2.0) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.9999999697559053) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + (y / t)) / (x + 1.0d0)
    t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_2 <= 0.9999999697559053d0) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 0.9999999697559053) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= 0.9999999697559053:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 0.9999999697559053)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 0.9999999697559053)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.9999999697559053], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999999969755905327 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 74.3

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

   4. Applied rewrites 74.3%

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

   if 0.999999969755905327 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.9%

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

Alternative 14: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 5e-11)
     (/ (+ x (/ y t)) 1.0)
     (if (<= t_1 2.0) 1.0 (+ 1.0 (/ y (* t (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e-11) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_1 <= 5d-11) then
        tmp = (x + (y / t)) / 1.0d0
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + (y / (t * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e-11) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= 5e-11:
		tmp = (x + (y / t)) / 1.0
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = 1.0 + (y / (t * (1.0 + x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= 5e-11)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(y / Float64(t * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= 5e-11)
		tmp = (x + (y / t)) / 1.0;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = 1.0 + (y / (t * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-11], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(1.0 + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-11

   1. Initial program 89.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 73.2%

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

   if 5.00000000000000018e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{1} \]

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 60.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 70.2

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

   7. Applied rewrites 70.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 63.9%

       \[\leadsto 1 + \frac{y}{\color{blue}{t} \cdot \left(1 + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 77.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.9e-32) 1.0 (if (<= x 1.16e-7) (/ (+ x (/ y t)) 1.0) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.9e-32) {
		tmp = 1.0;
	} else if (x <= 1.16e-7) {
		tmp = (x + (y / t)) / 1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.9d-32)) then
        tmp = 1.0d0
    else if (x <= 1.16d-7) then
        tmp = (x + (y / t)) / 1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.9e-32) {
		tmp = 1.0;
	} else if (x <= 1.16e-7) {
		tmp = (x + (y / t)) / 1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.9e-32:
		tmp = 1.0
	elif x <= 1.16e-7:
		tmp = (x + (y / t)) / 1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.9e-32)
		tmp = 1.0;
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.9e-32)
		tmp = 1.0;
	elseif (x <= 1.16e-7)
		tmp = (x + (y / t)) / 1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.9e-32], 1.0, If[LessEqual[x, 1.16e-7], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999996e-32 or 1.1600000000000001e-7 < x

   1. Initial program 88.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 86.3%

      \[\leadsto \color{blue}{1} \]

   if -2.89999999999999996e-32 < x < 1.1600000000000001e-7

   1. Initial program 90.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 68.1

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 67.9%

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

Alternative 16: 68.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.8e-45) 1.0 (if (<= x 3.5e-84) (/ y t) (/ x (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.8e-45) {
		tmp = 1.0;
	} else if (x <= 3.5e-84) {
		tmp = y / t;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-4.8d-45)) then
        tmp = 1.0d0
    else if (x <= 3.5d-84) then
        tmp = y / t
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.8e-45) {
		tmp = 1.0;
	} else if (x <= 3.5e-84) {
		tmp = y / t;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.8e-45:
		tmp = 1.0
	elif x <= 3.5e-84:
		tmp = y / t
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.8e-45)
		tmp = 1.0;
	elseif (x <= 3.5e-84)
		tmp = Float64(y / t);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.8e-45)
		tmp = 1.0;
	elseif (x <= 3.5e-84)
		tmp = y / t;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.8e-45], 1.0, If[LessEqual[x, 3.5e-84], N[(y / t), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999998e-45

   1. Initial program 88.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{1} \]

   if -4.7999999999999998e-45 < x < 3.5000000000000001e-84

   1. Initial program 90.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 50.9

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

   4. Applied rewrites 50.9%

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

   if 3.5000000000000001e-84 < x

   1. Initial program 89.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 77.4%

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

Alternative 17: 68.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.8e-45) 1.0 (if (<= x 3.1e-83) (/ y t) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.8e-45) {
		tmp = 1.0;
	} else if (x <= 3.1e-83) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-4.8d-45)) then
        tmp = 1.0d0
    else if (x <= 3.1d-83) then
        tmp = y / t
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.8e-45) {
		tmp = 1.0;
	} else if (x <= 3.1e-83) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.8e-45:
		tmp = 1.0
	elif x <= 3.1e-83:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.8e-45)
		tmp = 1.0;
	elseif (x <= 3.1e-83)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.8e-45)
		tmp = 1.0;
	elseif (x <= 3.1e-83)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.8e-45], 1.0, If[LessEqual[x, 3.1e-83], N[(y / t), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7999999999999998e-45 or 3.09999999999999992e-83 < x

   1. Initial program 89.2%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{1} \]

   if -4.7999999999999998e-45 < x < 3.09999999999999992e-83

   1. Initial program 90.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 50.9

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

   4. Applied rewrites 50.9%

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

Alternative 18: 54.0% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 1.0)

C

double code(double x, double y, double z, double t) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0;
}

Python

def code(x, y, z, t):
	return 1.0

Julia

function code(x, y, z, t)
	return 1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 89.6%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 54.0%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))