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

Percentage Accurate: 89.2% → 96.1%

Time: 4.5s

Alternatives: 14

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% 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: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x - -1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{+198}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{-1 - x} - \frac{x}{\left(-1 - x\right) \cdot z}}{t}, -1, \frac{x}{1 + x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (- x -1.0))))
   (if (<= t_2 -1e+249)
     (* (/ y (+ 1.0 x)) (/ z t_1))
     (if (<= t_2 1e+198)
       t_2
       (fma
        (/ (- (/ y (- -1.0 x)) (/ x (* (- -1.0 x) z))) t)
        -1.0
        (/ x (+ 1.0 x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_2 <= -1e+249) {
		tmp = (y / (1.0 + x)) * (z / t_1);
	} else if (t_2 <= 1e+198) {
		tmp = t_2;
	} else {
		tmp = fma((((y / (-1.0 - x)) - (x / ((-1.0 - x) * z))) / t), -1.0, (x / (1.0 + x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= -1e+249)
		tmp = Float64(Float64(y / Float64(1.0 + x)) * Float64(z / t_1));
	elseif (t_2 <= 1e+198)
		tmp = t_2;
	else
		tmp = fma(Float64(Float64(Float64(y / Float64(-1.0 - x)) - Float64(x / Float64(Float64(-1.0 - x) * z))) / t), -1.0, Float64(x / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+249], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+198], t$95$2, N[(N[(N[(N[(y / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(-1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * -1.0 + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999992e248

   1. Initial program 52.9%

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

   3. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e198

   1. Initial program 98.9%

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

   if 1.00000000000000002e198 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 17.8%

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

   3. Taylor expanded in t around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 10^{+198}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{-1 - x} - \frac{x}{\left(-1 - x\right) \cdot z}}{t}, -1, \frac{x}{1 + x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{y}{1 + x} \cdot \frac{z}{t\_1}\\ t_3 := x + \frac{y \cdot z - x}{t\_1}\\ t_4 := \frac{t\_3}{x - -1}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 10^{-9}:\\ \;\;\;\;\frac{t\_3}{1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (* (/ y (+ 1.0 x)) (/ z t_1)))
        (t_3 (+ x (/ (- (* y z) x) t_1)))
        (t_4 (/ t_3 (- x -1.0))))
   (if (<= t_4 -2e+40)
     t_2
     (if (<= t_4 1e-9)
       (/ t_3 1.0)
       (if (<= t_4 2.0)
         (/ (- x (/ x t_1)) (- x -1.0))
         (if (<= t_4 INFINITY) t_2 (/ (+ x (/ y t)) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (y / (1.0 + x)) * (z / t_1);
	double t_3 = x + (((y * z) - x) / t_1);
	double t_4 = t_3 / (x - -1.0);
	double tmp;
	if (t_4 <= -2e+40) {
		tmp = t_2;
	} else if (t_4 <= 1e-9) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (x - -1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (y / (1.0 + x)) * (z / t_1);
	double t_3 = x + (((y * z) - x) / t_1);
	double t_4 = t_3 / (x - -1.0);
	double tmp;
	if (t_4 <= -2e+40) {
		tmp = t_2;
	} else if (t_4 <= 1e-9) {
		tmp = t_3 / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (x - -1.0);
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (y / (1.0 + x)) * (z / t_1)
	t_3 = x + (((y * z) - x) / t_1)
	t_4 = t_3 / (x - -1.0)
	tmp = 0
	if t_4 <= -2e+40:
		tmp = t_2
	elif t_4 <= 1e-9:
		tmp = t_3 / 1.0
	elif t_4 <= 2.0:
		tmp = (x - (x / t_1)) / (x - -1.0)
	elif t_4 <= math.inf:
		tmp = t_2
	else:
		tmp = (x + (y / t)) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(y / Float64(1.0 + x)) * Float64(z / t_1))
	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+40)
		tmp = t_2;
	elseif (t_4 <= 1e-9)
		tmp = Float64(t_3 / 1.0);
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x - -1.0));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (y / (1.0 + x)) * (z / t_1);
	t_3 = x + (((y * z) - x) / t_1);
	t_4 = t_3 / (x - -1.0);
	tmp = 0.0;
	if (t_4 <= -2e+40)
		tmp = t_2;
	elseif (t_4 <= 1e-9)
		tmp = t_3 / 1.0;
	elseif (t_4 <= 2.0)
		tmp = (x - (x / t_1)) / (x - -1.0);
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = (x + (y / t)) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $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+40], t$95$2, If[LessEqual[t$95$4, 1e-9], N[(t$95$3 / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000006e40 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 67.7%

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

   3. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 92.8

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

   5. Applied rewrites 92.8%

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

   if -2.00000000000000006e40 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000006e-9

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.7%

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

   if 1.00000000000000006e-9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 99.1

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 10^{-9}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := t \cdot z - x\\ t_3 := \frac{y}{1 + x} \cdot \frac{z}{t\_2}\\ t_4 := \frac{x + \frac{t\_1}{t\_2}}{x - -1}\\ \mathbf{if}\;t\_4 \leq -20000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 10^{-9}:\\ \;\;\;\;\frac{x + \frac{t\_1}{t \cdot z}}{1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* t z) x))
        (t_3 (* (/ y (+ 1.0 x)) (/ z t_2)))
        (t_4 (/ (+ x (/ t_1 t_2)) (- x -1.0))))
   (if (<= t_4 -20000.0)
     t_3
     (if (<= t_4 1e-9)
       (/ (+ 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 (/ (+ x (/ y t)) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (t * z) - x;
	double t_3 = (y / (1.0 + x)) * (z / t_2);
	double t_4 = (x + (t_1 / t_2)) / (x - -1.0);
	double tmp;
	if (t_4 <= -20000.0) {
		tmp = t_3;
	} else if (t_4 <= 1e-9) {
		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 = (x + (y / t)) / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (t * z) - x;
	double t_3 = (y / (1.0 + x)) * (z / t_2);
	double t_4 = (x + (t_1 / t_2)) / (x - -1.0);
	double tmp;
	if (t_4 <= -20000.0) {
		tmp = t_3;
	} else if (t_4 <= 1e-9) {
		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 = (x + (y / t)) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) - x
	t_2 = (t * z) - x
	t_3 = (y / (1.0 + x)) * (z / t_2)
	t_4 = (x + (t_1 / t_2)) / (x - -1.0)
	tmp = 0
	if t_4 <= -20000.0:
		tmp = t_3
	elif t_4 <= 1e-9:
		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 = (x + (y / t)) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - x)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(y / Float64(1.0 + x)) * Float64(z / t_2))
	t_4 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_4 <= -20000.0)
		tmp = t_3;
	elseif (t_4 <= 1e-9)
		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(x + Float64(y / t)) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) - x;
	t_2 = (t * z) - x;
	t_3 = (y / (1.0 + x)) * (z / t_2);
	t_4 = (x + (t_1 / t_2)) / (x - -1.0);
	tmp = 0.0;
	if (t_4 <= -20000.0)
		tmp = t_3;
	elseif (t_4 <= 1e-9)
		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 = (x + (y / t)) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$2), $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, -20000.0], t$95$3, If[LessEqual[t$95$4, 1e-9], 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[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e4 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 69.3%

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

   3. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if -2e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000006e-9

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.4%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z}}}{1} \]
   10. 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. lower-*.f64 N/A

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

       4. lift-*.f64 92.6

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

   11. Applied rewrites 92.6%

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

   if 1.00000000000000006e-9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 99.1

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -20000:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 10^{-9}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x - -1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{y}{1 + x} \cdot \frac{z}{t\_2}\\ t_4 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x - -1}\\ \mathbf{if}\;t\_4 \leq -20000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (- x -1.0)))
        (t_2 (- (* t z) x))
        (t_3 (* (/ y (+ 1.0 x)) (/ z t_2)))
        (t_4 (/ (+ x (/ (- (* y z) x) t_2)) (- x -1.0))))
   (if (<= t_4 -20000.0)
     t_3
     (if (<= t_4 1e-9)
       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 = (x + (y / t)) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (y / (1.0 + x)) * (z / t_2);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x - -1.0);
	double tmp;
	if (t_4 <= -20000.0) {
		tmp = t_3;
	} else if (t_4 <= 1e-9) {
		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 = (x + (y / t)) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (y / (1.0 + x)) * (z / t_2);
	double t_4 = (x + (((y * z) - x) / t_2)) / (x - -1.0);
	double tmp;
	if (t_4 <= -20000.0) {
		tmp = t_3;
	} else if (t_4 <= 1e-9) {
		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 = (x + (y / t)) / (x - -1.0)
	t_2 = (t * z) - x
	t_3 = (y / (1.0 + x)) * (z / t_2)
	t_4 = (x + (((y * z) - x) / t_2)) / (x - -1.0)
	tmp = 0
	if t_4 <= -20000.0:
		tmp = t_3
	elif t_4 <= 1e-9:
		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(x + Float64(y / t)) / Float64(x - -1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(y / Float64(1.0 + x)) * Float64(z / 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 <= -20000.0)
		tmp = t_3;
	elseif (t_4 <= 1e-9)
		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 = (x + (y / t)) / (x - -1.0);
	t_2 = (t * z) - x;
	t_3 = (y / (1.0 + x)) * (z / t_2);
	t_4 = (x + (((y * z) - x) / t_2)) / (x - -1.0);
	tmp = 0.0;
	if (t_4 <= -20000.0)
		tmp = t_3;
	elseif (t_4 <= 1e-9)
		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[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -20000.0], t$95$3, If[LessEqual[t$95$4, 1e-9], 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))) < -2e4 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 69.3%

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

   3. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if -2e4 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000006e-9 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 80.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if 1.00000000000000006e-9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -20000:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 10^{-9}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t \cdot \left(1 + x\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-278}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.99999:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* t (+ 1.0 x))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (<= t_2 2e-278)
     t_1
     (if (<= t_2 0.99999) (/ x (- x -1.0)) (if (<= t_2 2.0) 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (1.0 + x));
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= 2e-278) {
		tmp = t_1;
	} else if (t_2 <= 0.99999) {
		tmp = x / (x - -1.0);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y / (t * (1.0d0 + x))
    t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_2 <= 2d-278) then
        tmp = t_1
    else if (t_2 <= 0.99999d0) then
        tmp = x / (x - (-1.0d0))
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (1.0 + x));
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_2 <= 2e-278) {
		tmp = t_1;
	} else if (t_2 <= 0.99999) {
		tmp = x / (x - -1.0);
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (t * (1.0 + x))
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_2 <= 2e-278:
		tmp = t_1
	elif t_2 <= 0.99999:
		tmp = x / (x - -1.0)
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(t * Float64(1.0 + x)))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= 2e-278)
		tmp = t_1;
	elseif (t_2 <= 0.99999)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / (t * (1.0 + x));
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= 2e-278)
		tmp = t_1;
	elseif (t_2 <= 0.99999)
		tmp = x / (x - -1.0);
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-278], t$95$1, If[LessEqual[t$95$2, 0.99999], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999988e-278 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 26.1

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 77.1

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in y around inf

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 54.5

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

   11. Applied rewrites 54.5%

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

   if 1.99999999999999988e-278 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999990000000000046

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 61.0%

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

   if 0.999990000000000046 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-278}:\\ \;\;\;\;\frac{y}{t \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 0.99999:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x - -1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{+198}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\right) - \frac{x}{t \cdot \left(z \cdot \left(1 + x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (- x -1.0))))
   (if (<= t_2 -1e+249)
     (* (/ y (+ 1.0 x)) (/ z t_1))
     (if (<= t_2 1e+198)
       t_2
       (-
        (+ (/ x (+ 1.0 x)) (/ y (* t (+ 1.0 x))))
        (/ x (* t (* z (+ 1.0 x)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_2 <= -1e+249) {
		tmp = (y / (1.0 + x)) * (z / t_1);
	} else if (t_2 <= 1e+198) {
		tmp = t_2;
	} else {
		tmp = ((x / (1.0 + x)) + (y / (t * (1.0 + x)))) - (x / (t * (z * (1.0 + x))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x - (-1.0d0))
    if (t_2 <= (-1d+249)) then
        tmp = (y / (1.0d0 + x)) * (z / t_1)
    else if (t_2 <= 1d+198) then
        tmp = t_2
    else
        tmp = ((x / (1.0d0 + x)) + (y / (t * (1.0d0 + x)))) - (x / (t * (z * (1.0d0 + x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_2 <= -1e+249) {
		tmp = (y / (1.0 + x)) * (z / t_1);
	} else if (t_2 <= 1e+198) {
		tmp = t_2;
	} else {
		tmp = ((x / (1.0 + x)) + (y / (t * (1.0 + x)))) - (x / (t * (z * (1.0 + x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0)
	tmp = 0
	if t_2 <= -1e+249:
		tmp = (y / (1.0 + x)) * (z / t_1)
	elif t_2 <= 1e+198:
		tmp = t_2
	else:
		tmp = ((x / (1.0 + x)) + (y / (t * (1.0 + x)))) - (x / (t * (z * (1.0 + x))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= -1e+249)
		tmp = Float64(Float64(y / Float64(1.0 + x)) * Float64(z / t_1));
	elseif (t_2 <= 1e+198)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(x / Float64(1.0 + x)) + Float64(y / Float64(t * Float64(1.0 + x)))) - Float64(x / Float64(t * Float64(z * Float64(1.0 + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= -1e+249)
		tmp = (y / (1.0 + x)) * (z / t_1);
	elseif (t_2 <= 1e+198)
		tmp = t_2;
	else
		tmp = ((x / (1.0 + x)) + (y / (t * (1.0 + x)))) - (x / (t * (z * (1.0 + x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+249], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+198], t$95$2, N[(N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(t * N[(z * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999992e248

   1. Initial program 52.9%

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

   3. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e198

   1. Initial program 98.9%

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

   if 1.00000000000000002e198 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 17.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 33.0

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

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 91.0

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

   8. Applied rewrites 91.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 10^{+198}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\right) - \frac{x}{t \cdot \left(z \cdot \left(1 + x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-278}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;t\_1 \leq 50000000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 2e-278)
     (/ y t)
     (if (<= t_1 0.1) (/ x 1.0) (if (<= t_1 50000000000000.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 2e-278) {
		tmp = y / t;
	} else if (t_1 <= 0.1) {
		tmp = x / 1.0;
	} else if (t_1 <= 50000000000000.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= 2d-278) then
        tmp = y / t
    else if (t_1 <= 0.1d0) then
        tmp = x / 1.0d0
    else if (t_1 <= 50000000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 2e-278) {
		tmp = y / t;
	} else if (t_1 <= 0.1) {
		tmp = x / 1.0;
	} else if (t_1 <= 50000000000000.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= 2e-278:
		tmp = y / t
	elif t_1 <= 0.1:
		tmp = x / 1.0
	elif t_1 <= 50000000000000.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 2e-278)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.1)
		tmp = Float64(x / 1.0);
	elseif (t_1 <= 50000000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= 2e-278)
		tmp = y / t;
	elseif (t_1 <= 0.1)
		tmp = x / 1.0;
	elseif (t_1 <= 50000000000000.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-278], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(x / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 50000000000000.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999988e-278 or 5e13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 47.5

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

   5. Applied rewrites 47.5%

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

   if 1.99999999999999988e-278 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 53.9%

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

   if 0.10000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-278}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 0.1:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 50000000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x - -1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{+198}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (- x -1.0))))
   (if (<= t_2 -1e+249)
     (* (/ y (+ 1.0 x)) (/ z t_1))
     (if (<= t_2 1e+198) t_2 (/ (+ x (/ y t)) (- x -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_2 <= -1e+249) {
		tmp = (y / (1.0 + x)) * (z / t_1);
	} else if (t_2 <= 1e+198) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x - (-1.0d0))
    if (t_2 <= (-1d+249)) then
        tmp = (y / (1.0d0 + x)) * (z / t_1)
    else if (t_2 <= 1d+198) then
        tmp = t_2
    else
        tmp = (x + (y / t)) / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_2 <= -1e+249) {
		tmp = (y / (1.0 + x)) * (z / t_1);
	} else if (t_2 <= 1e+198) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0)
	tmp = 0
	if t_2 <= -1e+249:
		tmp = (y / (1.0 + x)) * (z / t_1)
	elif t_2 <= 1e+198:
		tmp = t_2
	else:
		tmp = (x + (y / t)) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_2 <= -1e+249)
		tmp = Float64(Float64(y / Float64(1.0 + x)) * Float64(z / t_1));
	elseif (t_2 <= 1e+198)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	tmp = 0.0;
	if (t_2 <= -1e+249)
		tmp = (y / (1.0 + x)) * (z / t_1);
	elseif (t_2 <= 1e+198)
		tmp = t_2;
	else
		tmp = (x + (y / t)) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+249], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+198], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999992e248

   1. Initial program 52.9%

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

   3. Taylor expanded in y around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if -9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000002e198

   1. Initial program 98.9%

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

   if 1.00000000000000002e198 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 17.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 10^{+198}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x - -1}\\ \mathbf{if}\;t\_2 \leq 10^{-9} \lor \neg \left(t\_2 \leq 2\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (- x -1.0))))
   (if (or (<= t_2 1e-9) (not (<= t_2 2.0)))
     (/ (+ x (/ y t)) (- x -1.0))
     (/ (- x (/ x t_1)) (- x -1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0)) {
		tmp = (x + (y / t)) / (x - -1.0);
	} else {
		tmp = (x - (x / t_1)) / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * z) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x - (-1.0d0))
    if ((t_2 <= 1d-9) .or. (.not. (t_2 <= 2.0d0))) then
        tmp = (x + (y / t)) / (x - (-1.0d0))
    else
        tmp = (x - (x / t_1)) / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0)) {
		tmp = (x + (y / t)) / (x - -1.0);
	} else {
		tmp = (x - (x / t_1)) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0)
	tmp = 0
	if (t_2 <= 1e-9) or not (t_2 <= 2.0):
		tmp = (x + (y / t)) / (x - -1.0)
	else:
		tmp = (x - (x / t_1)) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x - -1.0))
	tmp = 0.0
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x - -1.0));
	else
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	tmp = 0.0;
	if ((t_2 <= 1e-9) || ~((t_2 <= 2.0)))
		tmp = (x + (y / t)) / (x - -1.0);
	else
		tmp = (x - (x / t_1)) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, 1e-9], N[Not[LessEqual[t$95$2, 2.0]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000006e-9 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if 1.00000000000000006e-9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 10^{-9} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{t \cdot z - x}}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 0.99999 \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (or (<= t_1 0.99999) (not (<= t_1 2.0)))
     (/ (+ x (/ y t)) (- x -1.0))
     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 <= 0.99999) || !(t_1 <= 2.0)) {
		tmp = (x + (y / t)) / (x - -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - (-1.0d0))
    if ((t_1 <= 0.99999d0) .or. (.not. (t_1 <= 2.0d0))) then
        tmp = (x + (y / t)) / (x - (-1.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if ((t_1 <= 0.99999) || !(t_1 <= 2.0)) {
		tmp = (x + (y / t)) / (x - -1.0);
	} else {
		tmp = 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 <= 0.99999) or not (t_1 <= 2.0):
		tmp = (x + (y / t)) / (x - -1.0)
	else:
		tmp = 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 <= 0.99999) || !(t_1 <= 2.0))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x - -1.0));
	else
		tmp = 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 <= 0.99999) || ~((t_1 <= 2.0)))
		tmp = (x + (y / t)) / (x - -1.0);
	else
		tmp = 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[Or[LessEqual[t$95$1, 0.99999], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.999990000000000046 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 76.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if 0.999990000000000046 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 0.99999 \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 0.1)
     (/ (+ x (/ y t)) 1.0)
     (if (<= t_1 2.0) 1.0 (/ y (* t (+ 1.0 x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / (t * (1.0 + x));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= 0.1d0) then
        tmp = (x + (y / t)) / 1.0d0
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / (t * (1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / (t * (1.0 + x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= 0.1:
		tmp = (x + (y / t)) / 1.0
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / (t * (1.0 + x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= 0.1)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / Float64(t * Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= 0.1)
		tmp = (x + (y / t)) / 1.0;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / (t * (1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(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, 0.1], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.4%

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

   if 0.10000000000000001 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.0%

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

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

   1. Initial program 41.4%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 81.1

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in y around inf

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 61.7

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

   11. Applied rewrites 61.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 0.1:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 10^{-9} \lor \neg \left(t\_1 \leq 50000000000000\right):\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (or (<= t_1 1e-9) (not (<= t_1 50000000000000.0))) (/ y t) 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 <= 1e-9) || !(t_1 <= 50000000000000.0)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - (-1.0d0))
    if ((t_1 <= 1d-9) .or. (.not. (t_1 <= 50000000000000.0d0))) then
        tmp = y / t
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if ((t_1 <= 1e-9) || !(t_1 <= 50000000000000.0)) {
		tmp = y / t;
	} else {
		tmp = 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 <= 1e-9) or not (t_1 <= 50000000000000.0):
		tmp = y / t
	else:
		tmp = 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 <= 1e-9) || !(t_1 <= 50000000000000.0))
		tmp = Float64(y / t);
	else
		tmp = 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 <= 1e-9) || ~((t_1 <= 50000000000000.0)))
		tmp = y / t;
	else
		tmp = 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[Or[LessEqual[t$95$1, 1e-9], N[Not[LessEqual[t$95$1, 50000000000000.0]], $MachinePrecision]], N[(y / t), $MachinePrecision], 1.0]]

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))) < 1.00000000000000006e-9 or 5e13 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 45.0

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

   5. Applied rewrites 45.0%

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

   if 1.00000000000000006e-9 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 10^{-9} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 50000000000000\right):\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.08e-33) 1.0 (if (<= x 2.7e-103) (/ y t) (/ x (- x -1.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.08e-33) {
		tmp = 1.0;
	} else if (x <= 2.7e-103) {
		tmp = y / t;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.08d-33)) then
        tmp = 1.0d0
    else if (x <= 2.7d-103) then
        tmp = y / t
    else
        tmp = x / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.08e-33) {
		tmp = 1.0;
	} else if (x <= 2.7e-103) {
		tmp = y / t;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.08e-33:
		tmp = 1.0
	elif x <= 2.7e-103:
		tmp = y / t
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.08e-33)
		tmp = 1.0;
	elseif (x <= 2.7e-103)
		tmp = Float64(y / t);
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.08e-33)
		tmp = 1.0;
	elseif (x <= 2.7e-103)
		tmp = y / t;
	else
		tmp = x / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.08e-33], 1.0, If[LessEqual[x, 2.7e-103], N[(y / t), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08000000000000007e-33

   1. Initial program 83.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 80.8%

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

   if -1.08000000000000007e-33 < x < 2.7000000000000001e-103

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 54.7

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

   5. Applied rewrites 54.7%

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

   if 2.7000000000000001e-103 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 83.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.9% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 50.2%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

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

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