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

Percentage Accurate: 89.1% → 97.3%

Time: 4.1s

Alternatives: 16

Speedup: 0.4×

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: 89.1% 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: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + y \cdot \frac{z}{t\_1}}{x + 1}\\ t_3 := x + \frac{y \cdot z - x}{t\_1}\\ t_4 := \frac{t\_3}{x + 1}\\ \mathbf{if}\;t\_4 \leq -50000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.9999999999970403:\\ \;\;\;\;\frac{t\_3}{1}\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

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 -50000000000.0)
     t_2
     (if (<= t_4 0.9999999999970403)
       (/ t_3 1.0)
       (if (<= t_4 1.0)
         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 <= -50000000000.0) {
		tmp = t_2;
	} else if (t_4 <= 0.9999999999970403) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 1.0) {
		tmp = 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 <= -50000000000.0) {
		tmp = t_2;
	} else if (t_4 <= 0.9999999999970403) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 1.0) {
		tmp = 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 <= -50000000000.0:
		tmp = t_2
	elif t_4 <= 0.9999999999970403:
		tmp = t_3 / 1.0
	elif t_4 <= 1.0:
		tmp = 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 <= -50000000000.0)
		tmp = t_2;
	elseif (t_4 <= 0.9999999999970403)
		tmp = Float64(t_3 / 1.0);
	elseif (t_4 <= 1.0)
		tmp = 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 <= -50000000000.0)
		tmp = t_2;
	elseif (t_4 <= 0.9999999999970403)
		tmp = t_3 / 1.0;
	elseif (t_4 <= 1.0)
		tmp = 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, -50000000000.0], t$95$2, If[LessEqual[t$95$4, 0.9999999999970403], N[(t$95$3 / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 1.0], 1.0, 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))) < -5e10 or 1 < (/.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 79.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   4. 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

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

      5. lift--.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if -5e10 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999999999997040256

   1. Initial program 95.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.7%

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

   if 0.999999999997040256 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.9%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 99.9

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

   5. 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: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + y \cdot \frac{z}{t\_2}}{x + 1}\\ t_4 := \frac{x + \frac{t\_1}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -50000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 10^{-8}:\\ \;\;\;\;\frac{x + \frac{t\_1}{t \cdot z}}{1}\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (* y (/ z t_2))) (+ x 1.0)))
        (t_4 (/ (+ x (/ t_1 t_2)) (+ x 1.0))))
   (if (<= t_4 -50000000000.0)
     t_3
     (if (<= t_4 1e-8)
       (/ (+ x (/ t_1 (* t z))) 1.0)
       (if (<= t_4 1.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_3 (/ (+ x (/ y t)) (+ x 1.0))))))))

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 + (y * (z / t_2))) / (x + 1.0);
	double t_4 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -50000000000.0) {
		tmp = t_3;
	} else if (t_4 <= 1e-8) {
		tmp = (x + (t_1 / (t * z))) / 1.0;
	} else if (t_4 <= 1.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} 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 = (y * z) - x;
	double t_2 = (t * z) - x;
	double t_3 = (x + (y * (z / t_2))) / (x + 1.0);
	double t_4 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -50000000000.0) {
		tmp = t_3;
	} else if (t_4 <= 1e-8) {
		tmp = (x + (t_1 / (t * z))) / 1.0;
	} else if (t_4 <= 1.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) - x
	t_2 = (t * z) - x
	t_3 = (x + (y * (z / t_2))) / (x + 1.0)
	t_4 = (x + (t_1 / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -50000000000.0:
		tmp = t_3
	elif t_4 <= 1e-8:
		tmp = (x + (t_1 / (t * z))) / 1.0
	elif t_4 <= 1.0:
		tmp = (x - (x / t_2)) / (x + 1.0)
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	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(y * Float64(z / t_2))) / Float64(x + 1.0))
	t_4 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -50000000000.0)
		tmp = t_3;
	elseif (t_4 <= 1e-8)
		tmp = Float64(Float64(x + Float64(t_1 / Float64(t * z))) / 1.0);
	elseif (t_4 <= 1.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_3;
	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 = (y * z) - x;
	t_2 = (t * z) - x;
	t_3 = (x + (y * (z / t_2))) / (x + 1.0);
	t_4 = (x + (t_1 / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -50000000000.0)
		tmp = t_3;
	elseif (t_4 <= 1e-8)
		tmp = (x + (t_1 / (t * z))) / 1.0;
	elseif (t_4 <= 1.0)
		tmp = (x - (x / t_2)) / (x + 1.0);
	elseif (t_4 <= Inf)
		tmp = t_3;
	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[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(y * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -50000000000.0], t$95$3, If[LessEqual[t$95$4, 1e-8], N[(N[(x + N[(t$95$1 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, 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))) < -5e10 or 1 < (/.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 79.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   4. 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

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

      5. lift--.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if -5e10 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-8

   1. Initial program 95.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in x around 0

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

      1. lift-*.f64 93.0

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

   8. Applied rewrites 93.0%

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

   if 1e-8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   4. 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

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

      4. lift--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 99.9

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

   5. 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: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;\frac{y \cdot z}{\left(1 + x\right) \cdot t\_2}\\ \mathbf{elif}\;t\_3 \leq 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x (+ 1.0 x)) (/ y (* t (+ 1.0 x)))))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 (- INFINITY))
     (/ (+ x (/ y t)) (+ x 1.0))
     (if (<= t_3 -4e+40)
       (/ (* y z) (* (+ 1.0 x) t_2))
       (if (<= t_3 1e-8) t_1 (if (<= t_3 1.0) 1.0 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else if (t_3 <= -4e+40) {
		tmp = (y * z) / ((1.0 + x) * t_2);
	} else if (t_3 <= 1e-8) {
		tmp = t_1;
	} else if (t_3 <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else if (t_3 <= -4e+40) {
		tmp = (y * z) / ((1.0 + x) * t_2);
	} else if (t_3 <= 1e-8) {
		tmp = t_1;
	} else if (t_3 <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / (1.0 + x)) + (y / (t * (1.0 + x)))
	t_2 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (x + (y / t)) / (x + 1.0)
	elif t_3 <= -4e+40:
		tmp = (y * z) / ((1.0 + x) * t_2)
	elif t_3 <= 1e-8:
		tmp = t_1
	elif t_3 <= 1.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(1.0 + x)) + Float64(y / Float64(t * Float64(1.0 + x))))
	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 <= Float64(-Inf))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	elseif (t_3 <= -4e+40)
		tmp = Float64(Float64(y * z) / Float64(Float64(1.0 + x) * t_2));
	elseif (t_3 <= 1e-8)
		tmp = t_1;
	elseif (t_3 <= 1.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	t_2 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (x + (y / t)) / (x + 1.0);
	elseif (t_3 <= -4e+40)
		tmp = (y * z) / ((1.0 + x) * t_2);
	elseif (t_3 <= 1e-8)
		tmp = t_1;
	elseif (t_3 <= 1.0)
		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[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $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, (-Infinity)], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e+40], N[(N[(y * z), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-8], t$95$1, If[LessEqual[t$95$3, 1.0], 1.0, t$95$1]]]]]]]

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))) < -inf.0

   1. Initial program 42.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000012e40

   1. Initial program 99.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if -4.00000000000000012e40 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-8 or 1 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 79.6%

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

   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. lower-*.f64 N/A

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

      3. lift-+.f64 77.4

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

   7. Applied rewrites 77.4%

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

   if 1e-8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.3%

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

Alternative 4: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;\frac{y \cdot z}{\left(1 + x\right) \cdot t\_2}\\ \mathbf{elif}\;t\_3 \leq 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0)))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -4e+40)
       (/ (* y z) (* (+ 1.0 x) t_2))
       (if (<= t_3 1e-8) t_1 (if (<= t_3 1.0) 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 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -4e+40) {
		tmp = (y * z) / ((1.0 + x) * t_2);
	} else if (t_3 <= 1e-8) {
		tmp = t_1;
	} else if (t_3 <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_3 <= -4e+40) {
		tmp = (y * z) / ((1.0 + x) * t_2);
	} else if (t_3 <= 1e-8) {
		tmp = t_1;
	} else if (t_3 <= 1.0) {
		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 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_1
	elif t_3 <= -4e+40:
		tmp = (y * z) / ((1.0 + x) * t_2)
	elif t_3 <= 1e-8:
		tmp = t_1
	elif t_3 <= 1.0:
		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(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 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -4e+40)
		tmp = Float64(Float64(y * z) / Float64(Float64(1.0 + x) * t_2));
	elseif (t_3 <= 1e-8)
		tmp = t_1;
	elseif (t_3 <= 1.0)
		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 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_1;
	elseif (t_3 <= -4e+40)
		tmp = (y * z) / ((1.0 + x) * t_2);
	elseif (t_3 <= 1e-8)
		tmp = t_1;
	elseif (t_3 <= 1.0)
		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[(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, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -4e+40], N[(N[(y * z), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-8], t$95$1, If[LessEqual[t$95$3, 1.0], 1.0, 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))) < -inf.0 or -4.00000000000000012e40 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-8 or 1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000012e40

   1. Initial program 99.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if 1e-8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.3%

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

Alternative 5: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{t\_1}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -50000000000:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 10^{-8}:\\ \;\;\;\;\frac{x + \frac{t\_1}{t \cdot z}}{1}\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

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 -50000000000.0)
     (* (/ y (+ 1.0 x)) (/ z t_2))
     (if (<= t_3 1e-8)
       (/ (+ x (/ t_1 (* t z))) 1.0)
       (if (<= t_3 1.0)
         (/ (- x (/ x t_2)) (+ 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 <= -50000000000.0) {
		tmp = (y / (1.0 + x)) * (z / t_2);
	} else if (t_3 <= 1e-8) {
		tmp = (x + (t_1 / (t * z))) / 1.0;
	} else if (t_3 <= 1.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} 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 = (y * z) - x
    t_2 = (t * z) - x
    t_3 = (x + (t_1 / t_2)) / (x + 1.0d0)
    if (t_3 <= (-50000000000.0d0)) then
        tmp = (y / (1.0d0 + x)) * (z / t_2)
    else if (t_3 <= 1d-8) then
        tmp = (x + (t_1 / (t * z))) / 1.0d0
    else if (t_3 <= 1.0d0) then
        tmp = (x - (x / t_2)) / (x + 1.0d0)
    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 = (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 <= -50000000000.0) {
		tmp = (y / (1.0 + x)) * (z / t_2);
	} else if (t_3 <= 1e-8) {
		tmp = (x + (t_1 / (t * z))) / 1.0;
	} else if (t_3 <= 1.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) - x
	t_2 = (t * z) - x
	t_3 = (x + (t_1 / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -50000000000.0:
		tmp = (y / (1.0 + x)) * (z / t_2)
	elif t_3 <= 1e-8:
		tmp = (x + (t_1 / (t * z))) / 1.0
	elif t_3 <= 1.0:
		tmp = (x - (x / t_2)) / (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 <= -50000000000.0)
		tmp = Float64(Float64(y / Float64(1.0 + x)) * Float64(z / t_2));
	elseif (t_3 <= 1e-8)
		tmp = Float64(Float64(x + Float64(t_1 / Float64(t * z))) / 1.0);
	elseif (t_3 <= 1.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / 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

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) - x;
	t_2 = (t * z) - x;
	t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -50000000000.0)
		tmp = (y / (1.0 + x)) * (z / t_2);
	elseif (t_3 <= 1e-8)
		tmp = (x + (t_1 / (t * z))) / 1.0;
	elseif (t_3 <= 1.0)
		tmp = (x - (x / t_2)) / (x + 1.0);
	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[(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, -50000000000.0], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-8], N[(N[(x + N[(t$95$1 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 1.0], N[(N[(x - N[(x / t$95$2), $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 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e10

   1. Initial program 77.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -5e10 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-8

   1. Initial program 95.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in x around 0

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

      1. lift-*.f64 93.0

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

   8. Applied rewrites 93.0%

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

   if 1e-8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   4. 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

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

      4. lift--.f64 99.4

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

   5. Applied rewrites 99.4%

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

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

   1. Initial program 61.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

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

   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. lower-*.f64 N/A

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

      3. lift-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

Alternative 6: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot z}{\left(1 + x\right) \cdot t\_1}\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 (- INFINITY))
     (/ (+ x (/ y t)) (+ x 1.0))
     (if (<= t_2 -5e-15)
       (/ (* y z) (* (+ 1.0 x) t_1))
       (if (<= t_2 1.0)
         (/ (- x (/ x t_1)) (+ 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 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else if (t_2 <= -5e-15) {
		tmp = (y * z) / ((1.0 + x) * t_1);
	} else if (t_2 <= 1.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	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) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else if (t_2 <= -5e-15) {
		tmp = (y * z) / ((1.0 + x) * t_1);
	} else if (t_2 <= 1.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} 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 <= -math.inf:
		tmp = (x + (y / t)) / (x + 1.0)
	elif t_2 <= -5e-15:
		tmp = (y * z) / ((1.0 + x) * t_1)
	elif t_2 <= 1.0:
		tmp = (x - (x / t_1)) / (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(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 <= Float64(-Inf))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	elseif (t_2 <= -5e-15)
		tmp = Float64(Float64(y * z) / Float64(Float64(1.0 + x) * t_1));
	elseif (t_2 <= 1.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / 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

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 <= -Inf)
		tmp = (x + (y / t)) / (x + 1.0);
	elseif (t_2 <= -5e-15)
		tmp = (y * z) / ((1.0 + x) * t_1);
	elseif (t_2 <= 1.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	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, (-Infinity)], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-15], N[(N[(y * z), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(N[(x - N[(x / t$95$1), $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 4 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 42.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.99999999999999999e-15

   1. Initial program 99.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 93.6

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

   7. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 93.6

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

   9. Applied rewrites 93.6%

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

   if -4.99999999999999999e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 98.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   4. 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

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

      4. lift--.f64 90.4

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

   5. Applied rewrites 90.4%

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

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

   1. Initial program 61.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

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

   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. lower-*.f64 N/A

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

      3. lift-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

Alternative 7: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{-8}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

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 -4e+40)
     (* y (/ z t_1))
     (if (<= t_2 1e-8)
       (/ (+ x (/ y t)) 1.0)
       (if (<= t_2 2.0) 1.0 (/ 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 <= -4e+40) {
		tmp = y * (z / t_1);
	} else if (t_2 <= 1e-8) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-4d+40)) then
        tmp = y * (z / t_1)
    else if (t_2 <= 1d-8) then
        tmp = (x + (y / t)) / 1.0d0
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = 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 <= -4e+40) {
		tmp = y * (z / t_1);
	} else if (t_2 <= 1e-8) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= -4e+40:
		tmp = y * (z / t_1)
	elif t_2 <= 1e-8:
		tmp = (x + (y / t)) / 1.0
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = 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 <= -4e+40)
		tmp = Float64(y * Float64(z / t_1));
	elseif (t_2 <= 1e-8)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = 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 <= -4e+40)
		tmp = y * (z / t_1);
	elseif (t_2 <= 1e-8)
		tmp = (x + (y / t)) / 1.0;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = 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, -4e+40], N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-8], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, N[(y / N[(t * N[(1.0 + x), $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))) < -4.00000000000000012e40

   1. Initial program 74.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 91.1

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 72.4%

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

   if -4.00000000000000012e40 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1e-8

   1. Initial program 95.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 82.1%

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

   if 1e-8 < (/.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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.7%

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

   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 59.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 71.2

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

   7. Applied rewrites 71.2%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-+.f64 57.8

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

   10. Applied rewrites 57.8%

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

Alternative 8: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;y \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

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 -5e-30)
     (* y (/ z t_1))
     (if (<= t_2 5e-26)
       (/ x (+ x 1.0))
       (if (<= t_2 2.0) 1.0 (/ 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 <= -5e-30) {
		tmp = y * (z / t_1);
	} else if (t_2 <= 5e-26) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-5d-30)) then
        tmp = y * (z / t_1)
    else if (t_2 <= 5d-26) then
        tmp = x / (x + 1.0d0)
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = 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 <= -5e-30) {
		tmp = y * (z / t_1);
	} else if (t_2 <= 5e-26) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= -5e-30:
		tmp = y * (z / t_1)
	elif t_2 <= 5e-26:
		tmp = x / (x + 1.0)
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = 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 <= -5e-30)
		tmp = Float64(y * Float64(z / t_1));
	elseif (t_2 <= 5e-26)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = 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 <= -5e-30)
		tmp = y * (z / t_1);
	elseif (t_2 <= 5e-26)
		tmp = x / (x + 1.0);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = 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, -5e-30], N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-26], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, N[(y / N[(t * N[(1.0 + x), $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))) < -4.99999999999999972e-30

   1. Initial program 80.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 69.6%

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

   if -4.99999999999999972e-30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000019e-26

   1. Initial program 94.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 53.4%

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

   if 5.00000000000000019e-26 < (/.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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.0%

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

   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 59.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 71.2

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

   7. Applied rewrites 71.2%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-+.f64 57.8

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

   10. Applied rewrites 57.8%

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

Alternative 9: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t \cdot \left(1 + x\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* t (+ 1.0 x))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 -2e-33)
     t_1
     (if (<= t_2 5e-26) (/ x (+ x 1.0)) (if (<= t_2 2.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (1.0 + x));
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e-33) {
		tmp = t_1;
	} else if (t_2 <= 5e-26) {
		tmp = x / (x + 1.0);
	} 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 = y / (t * (1.0d0 + x))
    t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_2 <= (-2d-33)) then
        tmp = t_1
    else if (t_2 <= 5d-26) then
        tmp = x / (x + 1.0d0)
    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 = y / (t * (1.0 + x));
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e-33) {
		tmp = t_1;
	} else if (t_2 <= 5e-26) {
		tmp = x / (x + 1.0);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (t * (1.0 + x))
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= -2e-33:
		tmp = t_1
	elif t_2 <= 5e-26:
		tmp = x / (x + 1.0)
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(t * Float64(1.0 + x)))
	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 <= -2e-33)
		tmp = t_1;
	elseif (t_2 <= 5e-26)
		tmp = Float64(x / Float64(x + 1.0));
	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 = y / (t * (1.0 + x));
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -2e-33)
		tmp = t_1;
	elseif (t_2 <= 5e-26)
		tmp = x / (x + 1.0);
	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[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $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, -2e-33], t$95$1, If[LessEqual[t$95$2, 5e-26], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, 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))) < -2.0000000000000001e-33 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 69.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 91.5

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 77.8

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

   7. Applied rewrites 77.8%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-+.f64 57.0

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

   10. Applied rewrites 57.0%

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

   if -2.0000000000000001e-33 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000019e-26

   1. Initial program 94.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 53.5%

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

   if 5.00000000000000019e-26 < (/.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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.0%

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

Alternative 10: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-33}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -2e-33)
     (/ y t)
     (if (<= t_1 5e-26) (/ x (+ x 1.0)) (if (<= t_1 2.0) 1.0 (/ y t))))))

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 <= -2e-33) {
		tmp = y / t;
	} else if (t_1 <= 5e-26) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	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 <= (-2d-33)) then
        tmp = y / t
    else if (t_1 <= 5d-26) then
        tmp = x / (x + 1.0d0)
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    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 <= -2e-33) {
		tmp = y / t;
	} else if (t_1 <= 5e-26) {
		tmp = x / (x + 1.0);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	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 <= -2e-33:
		tmp = y / t
	elif t_1 <= 5e-26:
		tmp = x / (x + 1.0)
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	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 <= -2e-33)
		tmp = Float64(y / t);
	elseif (t_1 <= 5e-26)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	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 <= -2e-33)
		tmp = y / t;
	elseif (t_1 <= 5e-26)
		tmp = x / (x + 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	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, -2e-33], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 5e-26], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $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))) < -2.0000000000000001e-33 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 69.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 50.6

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

   5. Applied rewrites 50.6%

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

   if -2.0000000000000001e-33 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000019e-26

   1. Initial program 94.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 53.5%

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

   if 5.00000000000000019e-26 < (/.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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.0%

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

Alternative 11: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+191}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

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 (- INFINITY))
     (/ (+ x (* y (/ z t_1))) (+ x 1.0))
     (if (<= t_2 5e+191) 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 <= -((double) INFINITY)) {
		tmp = (x + (y * (z / t_1))) / (x + 1.0);
	} else if (t_2 <= 5e+191) {
		tmp = t_2;
	} else {
		tmp = (x / (1.0 + x)) + (y / (t * (1.0 + x)));
	}
	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) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (x + (y * (z / t_1))) / (x + 1.0);
	} else if (t_2 <= 5e+191) {
		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 <= -math.inf:
		tmp = (x + (y * (z / t_1))) / (x + 1.0)
	elif t_2 <= 5e+191:
		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 <= Float64(-Inf))
		tmp = Float64(Float64(x + Float64(y * Float64(z / t_1))) / Float64(x + 1.0));
	elseif (t_2 <= 5e+191)
		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 <= -Inf)
		tmp = (x + (y * (z / t_1))) / (x + 1.0);
	elseif (t_2 <= 5e+191)
		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, (-Infinity)], N[(N[(x + N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+191], 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))) < -inf.0

   1. Initial program 42.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   4. 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

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

      5. lift--.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000002e191

   1. Initial program 98.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   if 5.0000000000000002e191 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 36.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 36.1%

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

   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. lower-*.f64 N/A

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

      3. lift-+.f64 80.3

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

   7. Applied rewrites 80.3%

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

Alternative 12: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -5e-15)
     (* (/ y (+ 1.0 x)) (/ z t_1))
     (if (<= t_2 1.0)
       (/ (- x (/ x t_1)) (+ 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 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e-15) {
		tmp = (y / (1.0 + x)) * (z / t_1);
	} else if (t_2 <= 1.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} 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 <= (-5d-15)) then
        tmp = (y / (1.0d0 + x)) * (z / t_1)
    else if (t_2 <= 1.0d0) then
        tmp = (x - (x / t_1)) / (x + 1.0d0)
    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 <= -5e-15) {
		tmp = (y / (1.0 + x)) * (z / t_1);
	} else if (t_2 <= 1.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} 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 <= -5e-15:
		tmp = (y / (1.0 + x)) * (z / t_1)
	elif t_2 <= 1.0:
		tmp = (x - (x / t_1)) / (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(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 <= -5e-15)
		tmp = Float64(Float64(y / Float64(1.0 + x)) * Float64(z / t_1));
	elseif (t_2 <= 1.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / 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

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 <= -5e-15)
		tmp = (y / (1.0 + x)) * (z / t_1);
	elseif (t_2 <= 1.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	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, -5e-15], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(N[(x - N[(x / t$95$1), $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))) < -4.99999999999999999e-15

   1. Initial program 79.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -4.99999999999999999e-15 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 98.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   4. 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

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

      4. lift--.f64 90.4

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

   5. Applied rewrites 90.4%

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

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

   1. Initial program 61.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

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

   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. lower-*.f64 N/A

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

      3. lift-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

Alternative 13: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 1e-8) t_1 (if (<= t_2 1.0) 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 <= 1e-8) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		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 <= 1d-8) then
        tmp = t_1
    else if (t_2 <= 1.0d0) 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 <= 1e-8) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		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 <= 1e-8:
		tmp = t_1
	elif t_2 <= 1.0:
		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 <= 1e-8)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		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 <= 1e-8)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		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, 1e-8], t$95$1, If[LessEqual[t$95$2, 1.0], 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))) < 1e-8 or 1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if 1e-8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.3%

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

Alternative 14: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 1e-8) (/ y t) (if (<= t_1 2.0) 1.0 (/ y t)))))

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 <= 1e-8) {
		tmp = y / t;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	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 <= 1d-8) then
        tmp = y / t
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    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 <= 1e-8) {
		tmp = y / t;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	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 <= 1e-8:
		tmp = y / t
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	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 <= 1e-8)
		tmp = Float64(y / t);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	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 <= 1e-8)
		tmp = y / t;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	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, 1e-8], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $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))) < 1e-8 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 78.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 45.5

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

   5. Applied rewrites 45.5%

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

   if 1e-8 < (/.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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.7%

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

Alternative 15: 98.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0)) <= Inf)
		tmp = Float64(Float64(fma(y, Float64(z / t_1), x) - Float64(x / t_1)) / Float64(x + 1.0));
	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]}, 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[(N[(y * N[(z / t$95$1), $MachinePrecision] + x), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $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 93.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate--l+ N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 98.5

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

   4. Applied rewrites 98.5%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 99.9

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

   5. 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 16: 54.1% accurate, 45.0× 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 89.1%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Applied rewrites 54.1%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025089 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

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

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))