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

Percentage Accurate: 89.8% → 95.6%

Time: 3.7s

Alternatives: 15

Speedup: 0.5×

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: 89.8% 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: 95.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{y \cdot \frac{z}{t\_1}}{x + 1}\\ t_3 := x + \frac{y \cdot z - x}{t\_1}\\ t_4 := \frac{t\_3}{x + 1}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_3}{1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (* y (/ z t_1)) (+ x 1.0)))
        (t_3 (+ x (/ (- (* y z) x) t_1)))
        (t_4 (/ t_3 (+ x 1.0))))
   (if (<= t_4 -2e+122)
     t_2
     (if (<= t_4 5e-11)
       (/ t_3 1.0)
       (if (<= t_4 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_2 (/ (+ (/ y t) x) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (y * (z / t_1)) / (x + 1.0);
	double t_3 = x + (((y * z) - x) / t_1);
	double t_4 = t_3 / (x + 1.0);
	double tmp;
	if (t_4 <= -2e+122) {
		tmp = t_2;
	} else if (t_4 <= 5e-11) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (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 = (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 <= -2e+122) {
		tmp = t_2;
	} else if (t_4 <= 5e-11) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (y * (z / t_1)) / (x + 1.0)
	t_3 = x + (((y * z) - x) / t_1)
	t_4 = t_3 / (x + 1.0)
	tmp = 0
	if t_4 <= -2e+122:
		tmp = t_2
	elif t_4 <= 5e-11:
		tmp = t_3 / 1.0
	elif t_4 <= 2.0:
		tmp = (x - (x / t_1)) / (x + 1.0)
	elif t_4 <= math.inf:
		tmp = t_2
	else:
		tmp = ((y / t) + x) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(y * Float64(z / t_1)) / Float64(x + 1.0))
	t_3 = Float64(x + Float64(Float64(Float64(y * z) - x) / t_1))
	t_4 = Float64(t_3 / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -2e+122)
		tmp = t_2;
	elseif (t_4 <= 5e-11)
		tmp = Float64(t_3 / 1.0);
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (y * (z / t_1)) / (x + 1.0);
	t_3 = x + (((y * z) - x) / t_1);
	t_4 = t_3 / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -2e+122)
		tmp = t_2;
	elseif (t_4 <= 5e-11)
		tmp = t_3 / 1.0;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_1)) / (x + 1.0);
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = ((y / t) + x) / (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[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+122], t$95$2, If[LessEqual[t$95$4, 5e-11], N[(t$95$3 / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(N[(N[(y / t), $MachinePrecision] + x), $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))) < -2.00000000000000003e122 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 31.9

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

   4. Applied rewrites 31.9%

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

   if -2.00000000000000003e122 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-11

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 45.1%

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

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

   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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

Alternative 2: 95.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := t \cdot z - x\\ t_3 := \frac{y \cdot \frac{z}{t\_2}}{x + 1}\\ t_4 := \frac{x + \frac{t\_1}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x + \frac{t\_1}{t \cdot z}}{1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* t z) x))
        (t_3 (/ (* y (/ z t_2)) (+ x 1.0)))
        (t_4 (/ (+ x (/ t_1 t_2)) (+ x 1.0))))
   (if (<= t_4 -2e+18)
     t_3
     (if (<= t_4 5e-11)
       (/ (+ x (/ t_1 (* t z))) 1.0)
       (if (<= t_4 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_3 (/ (+ (/ y t) x) (+ 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 = (y * (z / t_2)) / (x + 1.0);
	double t_4 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 5e-11) {
		tmp = (x + (t_1 / (t * z))) / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = ((y / t) + x) / (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 = (y * (z / t_2)) / (x + 1.0);
	double t_4 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 5e-11) {
		tmp = (x + (t_1 / (t * z))) / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) - x
	t_2 = (t * z) - x
	t_3 = (y * (z / t_2)) / (x + 1.0)
	t_4 = (x + (t_1 / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -2e+18:
		tmp = t_3
	elif t_4 <= 5e-11:
		tmp = (x + (t_1 / (t * z))) / 1.0
	elif t_4 <= 2.0:
		tmp = (x - (x / t_2)) / (x + 1.0)
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = ((y / t) + x) / (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(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 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 5e-11)
		tmp = Float64(Float64(x + Float64(t_1 / Float64(t * z))) / 1.0);
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / 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 = (y * (z / t_2)) / (x + 1.0);
	t_4 = (x + (t_1 / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 5e-11)
		tmp = (x + (t_1 / (t * z))) / 1.0;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_2)) / (x + 1.0);
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = ((y / t) + x) / (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[(y * N[(z / t$95$2), $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, -2e+18], t$95$3, If[LessEqual[t$95$4, 5e-11], N[(N[(x + N[(t$95$1 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 2.0], 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[(N[(y / t), $MachinePrecision] + x), $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))) < -2e18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 31.9

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

   4. Applied rewrites 31.9%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-11

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 45.1%

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

   5. Taylor expanded in x around 0

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

      1. lift-*.f64 35.0

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

   7. Applied rewrites 35.0%

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

   8. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 31.3

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

   10. Applied rewrites 31.3%

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

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

   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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

Alternative 3: 94.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[N[(N[(x + N[(t$95$1 / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x + N[(t$95$1 / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t), $MachinePrecision] + x), $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 89.8%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 89.8

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

   4. Applied rewrites 89.8%

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

   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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

Alternative 4: 93.5% accurate, 0.5× speedup

Math

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

FPCore

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

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 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

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 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	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 <= math.inf:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (x + 1.0)
	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 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	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 <= Inf)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (x + 1.0);
	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, Infinity], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $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 89.8%

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

   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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

Alternative 5: 93.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{y \cdot \frac{z}{t\_2}}{x + 1}\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (- (* t z) x))
        (t_3 (/ (* y (/ z t_2)) (+ x 1.0)))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -2e+18)
     t_3
     (if (<= t_4 5e-11)
       t_1
       (if (<= t_4 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (y * (z / t_2)) / (x + 1.0);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 5e-11) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (y * (z / t_2)) / (x + 1.0);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 5e-11) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x + 1.0)
	t_2 = (t * z) - x
	t_3 = (y * (z / t_2)) / (x + 1.0)
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -2e+18:
		tmp = t_3
	elif t_4 <= 5e-11:
		tmp = t_1
	elif t_4 <= 2.0:
		tmp = (x - (x / t_2)) / (x + 1.0)
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(y * Float64(z / t_2)) / Float64(x + 1.0))
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 5e-11)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x + 1.0);
	t_2 = (t * z) - x;
	t_3 = (y * (z / t_2)) / (x + 1.0);
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 5e-11)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_2)) / (x + 1.0);
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+18], t$95$3, If[LessEqual[t$95$4, 5e-11], t$95$1, If[LessEqual[t$95$4, 2.0], 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, 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))) < -2e18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 31.9

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

   4. Applied rewrites 31.9%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-11 or +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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

Alternative 6: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ t_4 := \left(1 + x\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \frac{y}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{z \cdot y}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0)))
        (t_4 (* (+ 1.0 x) t_2)))
   (if (<= t_3 -2e+18)
     (* z (/ y t_4))
     (if (<= t_3 5e-11)
       t_1
       (if (<= t_3 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_3 INFINITY) (/ (* z y) t_4) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (1.0 + x) * t_2;
	double tmp;
	if (t_3 <= -2e+18) {
		tmp = z * (y / t_4);
	} else if (t_3 <= 5e-11) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (z * y) / t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (1.0 + x) * t_2;
	double tmp;
	if (t_3 <= -2e+18) {
		tmp = z * (y / t_4);
	} else if (t_3 <= 5e-11) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (z * y) / t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x + 1.0)
	t_2 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	t_4 = (1.0 + x) * t_2
	tmp = 0
	if t_3 <= -2e+18:
		tmp = z * (y / t_4)
	elif t_3 <= 5e-11:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = (x - (x / t_2)) / (x + 1.0)
	elif t_3 <= math.inf:
		tmp = (z * y) / t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / 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))
	t_4 = Float64(Float64(1.0 + x) * t_2)
	tmp = 0.0
	if (t_3 <= -2e+18)
		tmp = Float64(z * Float64(y / t_4));
	elseif (t_3 <= 5e-11)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(z * y) / t_4);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x + 1.0);
	t_2 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	t_4 = (1.0 + x) * t_2;
	tmp = 0.0;
	if (t_3 <= -2e+18)
		tmp = z * (y / t_4);
	elseif (t_3 <= 5e-11)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_2)) / (x + 1.0);
	elseif (t_3 <= Inf)
		tmp = (z * y) / t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $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]}, Block[{t$95$4 = N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+18], N[(z * N[(y / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-11], t$95$1, If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(z * y), $MachinePrecision] / t$95$4), $MachinePrecision], 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))) < -2e18

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 28.2

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

   4. Applied rewrites 28.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 28.2

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

   6. Applied rewrites 28.2%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-11 or +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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 28.2

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

   4. Applied rewrites 28.2%

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

Alternative 7: 90.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ t_4 := \left(1 + x\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \frac{y}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - -1}{x + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{z \cdot y}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0)))
        (t_4 (* (+ 1.0 x) t_2)))
   (if (<= t_3 -2e+18)
     (* z (/ y t_4))
     (if (<= t_3 5e-11)
       t_1
       (if (<= t_3 2.0)
         (/ (- x -1.0) (+ x 1.0))
         (if (<= t_3 INFINITY) (/ (* z y) t_4) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (1.0 + x) * t_2;
	double tmp;
	if (t_3 <= -2e+18) {
		tmp = z * (y / t_4);
	} else if (t_3 <= 5e-11) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x - -1.0) / (x + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (z * y) / t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = (1.0 + x) * t_2;
	double tmp;
	if (t_3 <= -2e+18) {
		tmp = z * (y / t_4);
	} else if (t_3 <= 5e-11) {
		tmp = t_1;
	} else if (t_3 <= 2.0) {
		tmp = (x - -1.0) / (x + 1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (z * y) / t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x + 1.0)
	t_2 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	t_4 = (1.0 + x) * t_2
	tmp = 0
	if t_3 <= -2e+18:
		tmp = z * (y / t_4)
	elif t_3 <= 5e-11:
		tmp = t_1
	elif t_3 <= 2.0:
		tmp = (x - -1.0) / (x + 1.0)
	elif t_3 <= math.inf:
		tmp = (z * y) / t_4
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / 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))
	t_4 = Float64(Float64(1.0 + x) * t_2)
	tmp = 0.0
	if (t_3 <= -2e+18)
		tmp = Float64(z * Float64(y / t_4));
	elseif (t_3 <= 5e-11)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - -1.0) / Float64(x + 1.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(z * y) / t_4);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x + 1.0);
	t_2 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	t_4 = (1.0 + x) * t_2;
	tmp = 0.0;
	if (t_3 <= -2e+18)
		tmp = z * (y / t_4);
	elseif (t_3 <= 5e-11)
		tmp = t_1;
	elseif (t_3 <= 2.0)
		tmp = (x - -1.0) / (x + 1.0);
	elseif (t_3 <= Inf)
		tmp = (z * y) / t_4;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $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]}, Block[{t$95$4 = N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+18], N[(z * N[(y / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-11], t$95$1, If[LessEqual[t$95$3, 2.0], N[(N[(x - -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(z * y), $MachinePrecision] / t$95$4), $MachinePrecision], 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))) < -2e18

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 28.2

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

   4. Applied rewrites 28.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 28.2

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

   6. Applied rewrites 28.2%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-11 or +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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 54.2%

      \[\leadsto \frac{x - -1}{x + 1} \]

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 28.2

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

   4. Applied rewrites 28.2%

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

Alternative 8: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := t \cdot z - x\\ t_3 := z \cdot \frac{y}{\left(1 + x\right) \cdot t\_2}\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - -1}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (- (* t z) x))
        (t_3 (* z (/ y (* (+ 1.0 x) t_2))))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_4 -2e+18)
     t_3
     (if (<= t_4 5e-11)
       t_1
       (if (<= t_4 2.0)
         (/ (- x -1.0) (+ x 1.0))
         (if (<= t_4 INFINITY) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = z * (y / ((1.0 + x) * t_2));
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 5e-11) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - -1.0) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (t * z) - x;
	double t_3 = z * (y / ((1.0 + x) * t_2));
	double t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_4 <= -2e+18) {
		tmp = t_3;
	} else if (t_4 <= 5e-11) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = (x - -1.0) / (x + 1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x + 1.0)
	t_2 = (t * z) - x
	t_3 = z * (y / ((1.0 + x) * t_2))
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_4 <= -2e+18:
		tmp = t_3
	elif t_4 <= 5e-11:
		tmp = t_1
	elif t_4 <= 2.0:
		tmp = (x - -1.0) / (x + 1.0)
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(z * Float64(y / Float64(Float64(1.0 + x) * t_2)))
	t_4 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 5e-11)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - -1.0) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x + 1.0);
	t_2 = (t * z) - x;
	t_3 = z * (y / ((1.0 + x) * t_2));
	t_4 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_4 <= -2e+18)
		tmp = t_3;
	elseif (t_4 <= 5e-11)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = (x - -1.0) / (x + 1.0);
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(y / N[(N[(1.0 + x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+18], t$95$3, If[LessEqual[t$95$4, 5e-11], t$95$1, If[LessEqual[t$95$4, 2.0], N[(N[(x - -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e18 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < +inf.0

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 28.2

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

   4. Applied rewrites 28.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 28.2

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

   6. Applied rewrites 28.2%

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

   if -2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000018e-11 or +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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 54.2%

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

Alternative 9: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\frac{x - -1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 5e-11) t_1 (if (<= t_2 1.0) (/ (- x -1.0) (+ x 1.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 5e-11) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = (x - -1.0) / (x + 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) + x) / (x + 1.0d0)
    t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_2 <= 5d-11) then
        tmp = t_1
    else if (t_2 <= 1.0d0) then
        tmp = (x - (-1.0d0)) / (x + 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) + x) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 5e-11) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = (x - -1.0) / (x + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x + 1.0)
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= 5e-11:
		tmp = t_1
	elif t_2 <= 1.0:
		tmp = (x - -1.0) / (x + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / 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 <= 5e-11)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = Float64(Float64(x - -1.0) / Float64(x + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x + 1.0);
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 5e-11)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = (x - -1.0) / (x + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $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, 5e-11], t$95$1, If[LessEqual[t$95$2, 1.0], N[(N[(x - -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 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))) < 5.00000000000000018e-11 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 89.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

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

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 54.2%

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

Alternative 10: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t \cdot \left(1 + x\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x - -1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 5e-11) t_1 (if (<= t_2 2e+18) (/ (- x -1.0) (+ x 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 <= 5e-11) {
		tmp = t_1;
	} else if (t_2 <= 2e+18) {
		tmp = (x - -1.0) / (x + 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 <= 5d-11) then
        tmp = t_1
    else if (t_2 <= 2d+18) then
        tmp = (x - (-1.0d0)) / (x + 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 <= 5e-11) {
		tmp = t_1;
	} else if (t_2 <= 2e+18) {
		tmp = (x - -1.0) / (x + 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 <= 5e-11:
		tmp = t_1
	elif t_2 <= 2e+18:
		tmp = (x - -1.0) / (x + 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 <= 5e-11)
		tmp = t_1;
	elseif (t_2 <= 2e+18)
		tmp = Float64(Float64(x - -1.0) / Float64(x + 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 <= 5e-11)
		tmp = t_1;
	elseif (t_2 <= 2e+18)
		tmp = (x - -1.0) / (x + 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, 5e-11], t$95$1, If[LessEqual[t$95$2, 2e+18], N[(N[(x - -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 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))) < 5.00000000000000018e-11 or 2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 28.2

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

   4. Applied rewrites 28.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y}{\color{blue}{t \cdot \left(1 + x\right)}} \]
   6. 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 25.6

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

   7. Applied rewrites 25.6%

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

   if 5.00000000000000018e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e18

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 54.2%

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

Alternative 11: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x - -1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 5e-11)
     (/ y t)
     (if (<= t_1 2e+18) (/ (- x -1.0) (+ x 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 <= 5e-11) {
		tmp = y / t;
	} else if (t_1 <= 2e+18) {
		tmp = (x - -1.0) / (x + 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 <= 5d-11) then
        tmp = y / t
    else if (t_1 <= 2d+18) then
        tmp = (x - (-1.0d0)) / (x + 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 <= 5e-11) {
		tmp = y / t;
	} else if (t_1 <= 2e+18) {
		tmp = (x - -1.0) / (x + 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 <= 5e-11:
		tmp = y / t
	elif t_1 <= 2e+18:
		tmp = (x - -1.0) / (x + 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 <= 5e-11)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e+18)
		tmp = Float64(Float64(x - -1.0) / Float64(x + 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 <= 5e-11)
		tmp = y / t;
	elseif (t_1 <= 2e+18)
		tmp = (x - -1.0) / (x + 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, 5e-11], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e+18], N[(N[(x - -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 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))) < 5.00000000000000018e-11 or 2e18 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

   if 5.00000000000000018e-11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e18

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 54.2%

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

Alternative 12: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-89}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 x))))
   (if (<= x -2.8e-60) t_1 (if (<= x 1.35e-89) (/ y t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (1.0 + x);
	double tmp;
	if (x <= -2.8e-60) {
		tmp = t_1;
	} else if (x <= 1.35e-89) {
		tmp = y / t;
	} 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) :: tmp
    t_1 = x / (1.0d0 + x)
    if (x <= (-2.8d-60)) then
        tmp = t_1
    else if (x <= 1.35d-89) then
        tmp = y / t
    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 / (1.0 + x);
	double tmp;
	if (x <= -2.8e-60) {
		tmp = t_1;
	} else if (x <= 1.35e-89) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (1.0 + x)
	tmp = 0
	if x <= -2.8e-60:
		tmp = t_1
	elif x <= 1.35e-89:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -2.8e-60)
		tmp = t_1;
	elseif (x <= 1.35e-89)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -2.8e-60)
		tmp = t_1;
	elseif (x <= 1.35e-89)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e-60], t$95$1, If[LessEqual[x, 1.35e-89], N[(y / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000002e-60 or 1.34999999999999994e-89 < x

   1. Initial program 89.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.9

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

   4. Applied rewrites 56.9%

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

   if -2.8000000000000002e-60 < x < 1.34999999999999994e-89

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

Alternative 13: 66.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{1}{x}\\ \mathbf{if}\;x \leq -1.66 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.82:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ 1.0 x))))
   (if (<= x -1.66e-9) t_1 (if (<= x 0.82) (/ y t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (1.0 / x);
	double tmp;
	if (x <= -1.66e-9) {
		tmp = t_1;
	} else if (x <= 0.82) {
		tmp = y / t;
	} 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) :: tmp
    t_1 = 1.0d0 - (1.0d0 / x)
    if (x <= (-1.66d-9)) then
        tmp = t_1
    else if (x <= 0.82d0) then
        tmp = y / t
    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 = 1.0 - (1.0 / x);
	double tmp;
	if (x <= -1.66e-9) {
		tmp = t_1;
	} else if (x <= 0.82) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - (1.0 / x)
	tmp = 0
	if x <= -1.66e-9:
		tmp = t_1
	elif x <= 0.82:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(1.0 / x))
	tmp = 0.0
	if (x <= -1.66e-9)
		tmp = t_1;
	elseif (x <= 0.82)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - (1.0 / x);
	tmp = 0.0;
	if (x <= -1.66e-9)
		tmp = t_1;
	elseif (x <= 0.82)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.66e-9], t$95$1, If[LessEqual[x, 0.82], N[(y / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6600000000000001e-9 or 0.819999999999999951 < x

   1. Initial program 89.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.9

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

   4. Applied rewrites 56.9%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 46.4

         \[\leadsto 1 - \frac{1}{x} \]

   7. Applied rewrites 46.4%

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

   if -1.6600000000000001e-9 < x < 0.819999999999999951

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

Alternative 14: 23.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{y}{t} \end{array} \]

FPCore

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

C

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

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 = y / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 23.6

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

4. Applied rewrites 23.6%

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

Alternative 15: 3.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(-1.0 / x), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 56.9

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

4. Applied rewrites 56.9%

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

5. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. lift-/.f64 46.4

      \[\leadsto 1 - \frac{1}{x} \]

7. Applied rewrites 46.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{x} \]
9. Step-by-step derivation

   1. lower-/.f64 3.4

      \[\leadsto \frac{-1}{x} \]

10. Applied rewrites 3.4%

    \[\leadsto \frac{-1}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025138 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))