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

Percentage Accurate: 88.8% → 95.8%

Time: 5.0s

Alternatives: 17

Speedup: 0.3×

Specification

Math

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

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: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_2 := z \cdot t - x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\frac{z}{\left(x - -1\right) \cdot t\_2} \cdot y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1 \cdot \frac{y - \frac{x}{z}}{t} - x}{-1 - x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{-1 \cdot \left(1 + x\right)}{-1 - x}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{x - -1} \cdot \frac{y}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_2 (- (* z t) x)))
   (if (<= t_1 -5e+98)
     (* (/ z (* (- x -1.0) t_2)) y)
     (if (<= t_1 5e-19)
       (/ (- (* -1.0 (/ (- y (/ x z)) t)) x) (- -1.0 x))
       (if (<= t_1 2.0)
         (/ (* -1.0 (+ 1.0 x)) (- -1.0 x))
         (if (<= t_1 INFINITY)
           (* (/ z (- x -1.0)) (/ y t_2))
           (/ (- (/ -1.0 (/ t y)) x) (- -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 t_2 = (z * t) - x;
	double tmp;
	if (t_1 <= -5e+98) {
		tmp = (z / ((x - -1.0) * t_2)) * y;
	} else if (t_1 <= 5e-19) {
		tmp = ((-1.0 * ((y - (x / z)) / t)) - x) / (-1.0 - x);
	} else if (t_1 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / (x - -1.0)) * (y / t_2);
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_2 = (z * t) - x;
	double tmp;
	if (t_1 <= -5e+98) {
		tmp = (z / ((x - -1.0) * t_2)) * y;
	} else if (t_1 <= 5e-19) {
		tmp = ((-1.0 * ((y - (x / z)) / t)) - x) / (-1.0 - x);
	} else if (t_1 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (z / (x - -1.0)) * (y / t_2);
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	t_2 = (z * t) - x
	tmp = 0
	if t_1 <= -5e+98:
		tmp = (z / ((x - -1.0) * t_2)) * y
	elif t_1 <= 5e-19:
		tmp = ((-1.0 * ((y - (x / z)) / t)) - x) / (-1.0 - x)
	elif t_1 <= 2.0:
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x)
	elif t_1 <= math.inf:
		tmp = (z / (x - -1.0)) * (y / t_2)
	else:
		tmp = ((-1.0 / (t / y)) - x) / (-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))
	t_2 = Float64(Float64(z * t) - x)
	tmp = 0.0
	if (t_1 <= -5e+98)
		tmp = Float64(Float64(z / Float64(Float64(x - -1.0) * t_2)) * y);
	elseif (t_1 <= 5e-19)
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(y - Float64(x / z)) / t)) - x) / Float64(-1.0 - x));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(-1.0 * Float64(1.0 + x)) / Float64(-1.0 - x));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / Float64(x - -1.0)) * Float64(y / t_2));
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(t / y)) - x) / 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);
	t_2 = (z * t) - x;
	tmp = 0.0;
	if (t_1 <= -5e+98)
		tmp = (z / ((x - -1.0) * t_2)) * y;
	elseif (t_1 <= 5e-19)
		tmp = ((-1.0 * ((y - (x / z)) / t)) - x) / (-1.0 - x);
	elseif (t_1 <= 2.0)
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	elseif (t_1 <= Inf)
		tmp = (z / (x - -1.0)) * (y / t_2);
	else
		tmp = ((-1.0 / (t / y)) - x) / (-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]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+98], N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 5e-19], N[(N[(N[(-1.0 * N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(-1.0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[(t / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999998e98

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 32.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 32.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

   if -4.9999999999999998e98 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e-19

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in t around -inf

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 57.8

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

   11. Applied rewrites 57.8%

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

   if 5.0000000000000004e-19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 52.8

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

   11. Applied rewrites 52.8%

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

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

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 30.3

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 30.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 30.3

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

   6. Applied rewrites 30.3%

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

   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 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+241}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t\_1}, z, x\right) - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+171}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\right) - \frac{x}{t \cdot \left(z \cdot \left(1 + x\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -5e+241)
     (/ (- (fma (/ y t_1) z x) (/ x t_1)) (+ x 1.0))
     (if (<= t_2 2e+171)
       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 <= -5e+241) {
		tmp = (fma((y / t_1), z, x) - (x / t_1)) / (x + 1.0);
	} else if (t_2 <= 2e+171) {
		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 <= -5e+241)
		tmp = Float64(Float64(fma(Float64(y / t_1), z, x) - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_2 <= 2e+171)
		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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+241], N[(N[(N[(N[(y / t$95$1), $MachinePrecision] * z + x), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+171], 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))) < -5.00000000000000025e241

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   if -5.00000000000000025e241 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e171

   1. Initial program 88.8%

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

   if 1.99999999999999991e171 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}\right) - \frac{x}{t \cdot \left(z \cdot \left(1 + x\right)\right)}} \]
   3. 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 60.7

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

   4. Applied rewrites 60.7%

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

Alternative 3: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+241}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t\_1}, z, x\right) - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+171}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -5e+241)
     (/ (- (fma (/ y t_1) z x) (/ x t_1)) (+ x 1.0))
     (if (<= t_2 2e+171) t_2 (/ (+ (/ y t) x) (- x -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+241) {
		tmp = (fma((y / t_1), z, x) - (x / t_1)) / (x + 1.0);
	} else if (t_2 <= 2e+171) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e+241)
		tmp = Float64(Float64(fma(Float64(y / t_1), z, x) - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_2 <= 2e+171)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+241], N[(N[(N[(N[(y / t$95$1), $MachinePrecision] * z + x), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+171], t$95$2, N[(N[(N[(y / t), $MachinePrecision] + x), $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))) < -5.00000000000000025e241

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   if -5.00000000000000025e241 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e171

   1. Initial program 88.8%

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

   if 1.99999999999999991e171 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 70.5

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

   6. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 70.5

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

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.5%

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

Alternative 4: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\frac{z}{\left(x - -1\right) \cdot \left(z \cdot t - x\right)} \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+171}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -4e+124)
     (* (/ z (* (- x -1.0) (- (* z t) x))) y)
     (if (<= t_1 2e+171) t_1 (/ (+ (/ y t) x) (- x -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -4e+124) {
		tmp = (z / ((x - -1.0) * ((z * t) - x))) * y;
	} else if (t_1 <= 2e+171) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + 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) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_1 <= (-4d+124)) then
        tmp = (z / ((x - (-1.0d0)) * ((z * t) - x))) * y
    else if (t_1 <= 2d+171) then
        tmp = t_1
    else
        tmp = ((y / t) + x) / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -4e+124) {
		tmp = (z / ((x - -1.0) * ((z * t) - x))) * y;
	} else if (t_1 <= 2e+171) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -4e+124:
		tmp = (z / ((x - -1.0) * ((z * t) - x))) * y
	elif t_1 <= 2e+171:
		tmp = t_1
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -4e+124)
		tmp = Float64(Float64(z / Float64(Float64(x - -1.0) * Float64(Float64(z * t) - x))) * y);
	elseif (t_1 <= 2e+171)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -4e+124)
		tmp = (z / ((x - -1.0) * ((z * t) - x))) * y;
	elseif (t_1 <= 2e+171)
		tmp = t_1;
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+124], N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+171], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $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))) < -3.99999999999999979e124

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 32.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 32.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

   if -3.99999999999999979e124 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e171

   1. Initial program 88.8%

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

   if 1.99999999999999991e171 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 70.5

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

   6. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 70.5

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

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.5%

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

Alternative 5: 93.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := x + \frac{y \cdot z - x}{t \cdot z - x}\\ t_2 := \frac{t\_1}{x + 1}\\ t_3 := z \cdot t - x\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\frac{z}{\left(x - -1\right) \cdot t\_3} \cdot y\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{-1 \cdot \left(1 + x\right)}{-1 - x}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{z}{x - -1} \cdot \frac{y}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (- (* y z) x) (- (* t z) x))))
        (t_2 (/ t_1 (+ x 1.0)))
        (t_3 (- (* z t) x)))
   (if (<= t_2 -4e+124)
     (* (/ z (* (- x -1.0) t_3)) y)
     (if (<= t_2 5e-19)
       (/ t_1 1.0)
       (if (<= t_2 2.0)
         (/ (* -1.0 (+ 1.0 x)) (- -1.0 x))
         (if (<= t_2 INFINITY)
           (* (/ z (- x -1.0)) (/ y t_3))
           (/ (- (/ -1.0 (/ t y)) x) (- -1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (((y * z) - x) / ((t * z) - x));
	double t_2 = t_1 / (x + 1.0);
	double t_3 = (z * t) - x;
	double tmp;
	if (t_2 <= -4e+124) {
		tmp = (z / ((x - -1.0) * t_3)) * y;
	} else if (t_2 <= 5e-19) {
		tmp = t_1 / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (z / (x - -1.0)) * (y / t_3);
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (((y * z) - x) / ((t * z) - x));
	double t_2 = t_1 / (x + 1.0);
	double t_3 = (z * t) - x;
	double tmp;
	if (t_2 <= -4e+124) {
		tmp = (z / ((x - -1.0) * t_3)) * y;
	} else if (t_2 <= 5e-19) {
		tmp = t_1 / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (z / (x - -1.0)) * (y / t_3);
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (((y * z) - x) / ((t * z) - x))
	t_2 = t_1 / (x + 1.0)
	t_3 = (z * t) - x
	tmp = 0
	if t_2 <= -4e+124:
		tmp = (z / ((x - -1.0) * t_3)) * y
	elif t_2 <= 5e-19:
		tmp = t_1 / 1.0
	elif t_2 <= 2.0:
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x)
	elif t_2 <= math.inf:
		tmp = (z / (x - -1.0)) * (y / t_3)
	else:
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x)))
	t_2 = Float64(t_1 / Float64(x + 1.0))
	t_3 = Float64(Float64(z * t) - x)
	tmp = 0.0
	if (t_2 <= -4e+124)
		tmp = Float64(Float64(z / Float64(Float64(x - -1.0) * t_3)) * y);
	elseif (t_2 <= 5e-19)
		tmp = Float64(t_1 / 1.0);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(-1.0 * Float64(1.0 + x)) / Float64(-1.0 - x));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(z / Float64(x - -1.0)) * Float64(y / t_3));
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(t / y)) - x) / 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));
	t_2 = t_1 / (x + 1.0);
	t_3 = (z * t) - x;
	tmp = 0.0;
	if (t_2 <= -4e+124)
		tmp = (z / ((x - -1.0) * t_3)) * y;
	elseif (t_2 <= 5e-19)
		tmp = t_1 / 1.0;
	elseif (t_2 <= 2.0)
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	elseif (t_2 <= Inf)
		tmp = (z / (x - -1.0)) * (y / t_3);
	else
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+124], N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, 5e-19], N[(t$95$1 / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(-1.0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(z / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[(t / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -3.99999999999999979e124

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 32.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 32.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

   if -3.99999999999999979e124 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e-19

   1. Initial program 88.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 46.1%

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

   if 5.0000000000000004e-19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 52.8

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

   11. Applied rewrites 52.8%

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

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

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 30.3

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 30.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 30.3

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

   6. Applied rewrites 30.3%

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

   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 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := z \cdot t - x\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\frac{z}{\left(x - -1\right) \cdot t\_1} \cdot y\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{z}{x - -1} \cdot \frac{y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 -5e+98)
     (* (/ z (* (- x -1.0) t_1)) y)
     (if (<= t_3 5e-89)
       (/ (+ (/ y t) x) (- x -1.0))
       (if (<= t_3 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_3 INFINITY)
           (* (/ z (- x -1.0)) (/ y t_1))
           (/ (- (/ -1.0 (/ t y)) x) (- -1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e+98) {
		tmp = (z / ((x - -1.0) * t_1)) * y;
	} else if (t_3 <= 5e-89) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (z / (x - -1.0)) * (y / t_1);
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e+98) {
		tmp = (z / ((x - -1.0) * t_1)) * y;
	} else if (t_3 <= 5e-89) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (z / (x - -1.0)) * (y / t_1);
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -5e+98:
		tmp = (z / ((x - -1.0) * t_1)) * y
	elif t_3 <= 5e-89:
		tmp = ((y / t) + x) / (x - -1.0)
	elif t_3 <= 2.0:
		tmp = (x - (x / t_2)) / (x + 1.0)
	elif t_3 <= math.inf:
		tmp = (z / (x - -1.0)) * (y / t_1)
	else:
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -5e+98)
		tmp = Float64(Float64(z / Float64(Float64(x - -1.0) * t_1)) * y);
	elseif (t_3 <= 5e-89)
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(z / Float64(x - -1.0)) * Float64(y / t_1));
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(t / y)) - x) / Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -5e+98)
		tmp = (z / ((x - -1.0) * t_1)) * y;
	elseif (t_3 <= 5e-89)
		tmp = ((y / t) + x) / (x - -1.0);
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_2)) / (x + 1.0);
	elseif (t_3 <= Inf)
		tmp = (z / (x - -1.0)) * (y / t_1);
	else
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+98], N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$3, 5e-89], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(z / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[(t / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999998e98

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 32.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 32.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

   if -4.9999999999999998e98 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999967e-89

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 70.5

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

   6. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 70.5

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

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.5%

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

   if 4.99999999999999967e-89 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

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

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 30.3

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 30.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 30.3

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

   6. Applied rewrites 30.3%

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

   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 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{z}{\left(x - -1\right) \cdot \left(z \cdot t - x\right)} \cdot y\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (* (- x -1.0) (- (* z t) x))) y))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 -5e+98)
     t_1
     (if (<= t_3 5e-89)
       (/ (+ (/ y t) x) (- x -1.0))
       (if (<= t_3 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_3 INFINITY) t_1 (/ (- (/ -1.0 (/ t y)) x) (- -1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / ((x - -1.0) * ((z * t) - x))) * y;
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e+98) {
		tmp = t_1;
	} else if (t_3 <= 5e-89) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / ((x - -1.0) * ((z * t) - x))) * y;
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e+98) {
		tmp = t_1;
	} else if (t_3 <= 5e-89) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / ((x - -1.0) * ((z * t) - x))) * y
	t_2 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -5e+98:
		tmp = t_1
	elif t_3 <= 5e-89:
		tmp = ((y / t) + x) / (x - -1.0)
	elif t_3 <= 2.0:
		tmp = (x - (x / t_2)) / (x + 1.0)
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(Float64(x - -1.0) * Float64(Float64(z * t) - x))) * y)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -5e+98)
		tmp = t_1;
	elseif (t_3 <= 5e-89)
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(t / y)) - x) / Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / ((x - -1.0) * ((z * t) - x))) * y;
	t_2 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -5e+98)
		tmp = t_1;
	elseif (t_3 <= 5e-89)
		tmp = ((y / t) + x) / (x - -1.0);
	elseif (t_3 <= 2.0)
		tmp = (x - (x / t_2)) / (x + 1.0);
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+98], t$95$1, If[LessEqual[t$95$3, 5e-89], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(N[(N[(-1.0 / N[(t / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999998e98 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 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 32.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 32.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

   if -4.9999999999999998e98 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999967e-89

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 70.5

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

   6. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 70.5

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

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.5%

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

   if 4.99999999999999967e-89 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 65.8

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

   4. Applied rewrites 65.8%

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

   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 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

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

Alternative 8: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{z}{\left(x - -1\right) \cdot \left(z \cdot t - x\right)} \cdot y\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{-1 \cdot \left(1 + x\right)}{-1 - x}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (* (- x -1.0) (- (* z t) x))) y))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 -5e+98)
     t_1
     (if (<= t_3 5e-89)
       (/ (+ (/ y t) x) (- x -1.0))
       (if (<= t_3 5e-19)
         (/ (- x (/ x t_2)) 1.0)
         (if (<= t_3 2.0)
           (/ (* -1.0 (+ 1.0 x)) (- -1.0 x))
           (if (<= t_3 INFINITY)
             t_1
             (/ (- (/ -1.0 (/ t y)) x) (- -1.0 x)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / ((x - -1.0) * ((z * t) - x))) * y;
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e+98) {
		tmp = t_1;
	} else if (t_3 <= 5e-89) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else if (t_3 <= 5e-19) {
		tmp = (x - (x / t_2)) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / ((x - -1.0) * ((z * t) - x))) * y;
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -5e+98) {
		tmp = t_1;
	} else if (t_3 <= 5e-89) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else if (t_3 <= 5e-19) {
		tmp = (x - (x / t_2)) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / ((x - -1.0) * ((z * t) - x))) * y
	t_2 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -5e+98:
		tmp = t_1
	elif t_3 <= 5e-89:
		tmp = ((y / t) + x) / (x - -1.0)
	elif t_3 <= 5e-19:
		tmp = (x - (x / t_2)) / 1.0
	elif t_3 <= 2.0:
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x)
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(Float64(x - -1.0) * Float64(Float64(z * t) - x))) * y)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -5e+98)
		tmp = t_1;
	elseif (t_3 <= 5e-89)
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	elseif (t_3 <= 5e-19)
		tmp = Float64(Float64(x - Float64(x / t_2)) / 1.0);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(-1.0 * Float64(1.0 + x)) / Float64(-1.0 - x));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(t / y)) - x) / Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / ((x - -1.0) * ((z * t) - x))) * y;
	t_2 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -5e+98)
		tmp = t_1;
	elseif (t_3 <= 5e-89)
		tmp = ((y / t) + x) / (x - -1.0);
	elseif (t_3 <= 5e-19)
		tmp = (x - (x / t_2)) / 1.0;
	elseif (t_3 <= 2.0)
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+98], t$95$1, If[LessEqual[t$95$3, 5e-89], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-19], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(-1.0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(N[(N[(-1.0 / N[(t / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999998e98 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 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 32.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 32.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

   if -4.9999999999999998e98 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999967e-89

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 70.5

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

   6. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 70.5

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

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.5%

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

   if 4.99999999999999967e-89 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e-19

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in y around 0

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

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

      4. lower-*.f64 65.8

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

   6. Applied rewrites 65.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 22.7%

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

   if 5.0000000000000004e-19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 52.8

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

   11. Applied rewrites 52.8%

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

   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 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{1}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{-1 \cdot \left(1 + x\right)}{-1 - x}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{\left(x - -1\right) \cdot \left(z \cdot t - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 5e-89)
     (/ (+ (/ y t) x) (- x -1.0))
     (if (<= t_2 5e-19)
       (/ (- x (/ x t_1)) 1.0)
       (if (<= t_2 2.0)
         (/ (* -1.0 (+ 1.0 x)) (- -1.0 x))
         (if (<= t_2 INFINITY)
           (* z (/ y (* (- x -1.0) (- (* z t) x))))
           (/ (- (/ -1.0 (/ t y)) 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 <= 5e-89) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else if (t_2 <= 5e-19) {
		tmp = (x - (x / t_1)) / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = z * (y / ((x - -1.0) * ((z * t) - x)));
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= 5e-89) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else if (t_2 <= 5e-19) {
		tmp = (x - (x / t_1)) / 1.0;
	} else if (t_2 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = z * (y / ((x - -1.0) * ((z * t) - x)));
	} else {
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= 5e-89:
		tmp = ((y / t) + x) / (x - -1.0)
	elif t_2 <= 5e-19:
		tmp = (x - (x / t_1)) / 1.0
	elif t_2 <= 2.0:
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x)
	elif t_2 <= math.inf:
		tmp = z * (y / ((x - -1.0) * ((z * t) - x)))
	else:
		tmp = ((-1.0 / (t / y)) - 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 <= 5e-89)
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	elseif (t_2 <= 5e-19)
		tmp = Float64(Float64(x - Float64(x / t_1)) / 1.0);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(-1.0 * Float64(1.0 + x)) / Float64(-1.0 - x));
	elseif (t_2 <= Inf)
		tmp = Float64(z * Float64(y / Float64(Float64(x - -1.0) * Float64(Float64(z * t) - x))));
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(t / y)) - x) / Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 5e-89)
		tmp = ((y / t) + x) / (x - -1.0);
	elseif (t_2 <= 5e-19)
		tmp = (x - (x / t_1)) / 1.0;
	elseif (t_2 <= 2.0)
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	elseif (t_2 <= Inf)
		tmp = z * (y / ((x - -1.0) * ((z * t) - x)));
	else
		tmp = ((-1.0 / (t / y)) - x) / (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-89], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-19], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(-1.0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(z * N[(y / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[(t / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999967e-89

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 70.5

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

   6. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 70.5

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

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.5%

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

   if 4.99999999999999967e-89 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e-19

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in y around 0

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

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

      4. lower-*.f64 65.8

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

   6. Applied rewrites 65.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 22.7%

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

   if 5.0000000000000004e-19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 52.8

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

   11. Applied rewrites 52.8%

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

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

   1. Initial program 88.8%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 29.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 29.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 29.4

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

   6. Applied rewrites 29.4%

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

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

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 88.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. sub-negate N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{t}{y}}}\right)\right) - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      9. distribute-neg-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{t}{y}}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{-1}}{\frac{t}{y}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{\frac{t}{y}}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-1}{\frac{t}{y}} - x}{\mathsf{neg}\left(\color{blue}{\left(x + 1\right)}\right)} \]

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x - -1}\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{-1 \cdot \left(1 + x\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (- x -1.0)))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0))))
   (if (<= t_3 5e-89)
     t_1
     (if (<= t_3 5e-19)
       (/ (- x (/ x t_2)) 1.0)
       (if (<= t_3 2.0) (/ (* -1.0 (+ 1.0 x)) (- -1.0 x)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= 5e-89) {
		tmp = t_1;
	} else if (t_3 <= 5e-19) {
		tmp = (x - (x / t_2)) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ((y / t) + x) / (x - (-1.0d0))
    t_2 = (t * z) - x
    t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0d0)
    if (t_3 <= 5d-89) then
        tmp = t_1
    else if (t_3 <= 5d-19) then
        tmp = (x - (x / t_2)) / 1.0d0
    else if (t_3 <= 2.0d0) then
        tmp = ((-1.0d0) * (1.0d0 + x)) / ((-1.0d0) - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (t * z) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= 5e-89) {
		tmp = t_1;
	} else if (t_3 <= 5e-19) {
		tmp = (x - (x / t_2)) / 1.0;
	} else if (t_3 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x - -1.0)
	t_2 = (t * z) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= 5e-89:
		tmp = t_1
	elif t_3 <= 5e-19:
		tmp = (x - (x / t_2)) / 1.0
	elif t_3 <= 2.0:
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= 5e-89)
		tmp = t_1;
	elseif (t_3 <= 5e-19)
		tmp = Float64(Float64(x - Float64(x / t_2)) / 1.0);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(-1.0 * Float64(1.0 + x)) / Float64(-1.0 - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x - -1.0);
	t_2 = (t * z) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= 5e-89)
		tmp = t_1;
	elseif (t_3 <= 5e-19)
		tmp = (x - (x / t_2)) / 1.0;
	elseif (t_3 <= 2.0)
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-89], t$95$1, If[LessEqual[t$95$3, 5e-19], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(-1.0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999967e-89 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 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.5

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.5%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x + 1} \]

      3. lower-+.f64 70.5

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x + 1} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{t} + x}{x - -1}} \]

   if 4.99999999999999967e-89 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e-19

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]
   5. 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. lower--.f64 N/A

         \[\leadsto \frac{x - \frac{x}{t \cdot z - \color{blue}{x}}}{x + 1} \]

      4. lower-*.f64 65.8

         \[\leadsto \frac{x - \frac{x}{t \cdot z - x}}{x + 1} \]

   6. Applied rewrites 65.8%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{x - \frac{x}{t \cdot z - x}}{\color{blue}{1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.7%

      \[\leadsto \frac{x - \frac{x}{t \cdot z - x}}{\color{blue}{1}} \]

   if 5.0000000000000004e-19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. div-flip N/A

         \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      4. lower-unsound-/.f64 88.7

         \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{y \cdot z} - x}}}{x + 1} \]

      6. *-commutative N/A

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{z \cdot y} - x}}}{x + 1} \]

      7. lower-*.f64 88.7

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{z \cdot y} - x}}}{x + 1} \]

   3. Applied rewrites 88.7%

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{z \cdot y - x}}}}{x + 1} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t}{y}}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-/.f64 70.4

         \[\leadsto \frac{x + \frac{1}{\frac{t}{\color{blue}{y}}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t}{y}}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{1}{\frac{t}{y}}}{x + 1}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(x + \frac{1}{\frac{t}{y}}\right)\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(x + \frac{1}{\frac{t}{y}}\right)\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x + \frac{1}{\frac{t}{y}}\right)}\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      5. add-flip N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x - \left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      6. sub-negate N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right) - x}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right) - x}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{t}{y}}}\right)\right) - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      9. distribute-neg-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{t}{y}}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{-1}}{\frac{t}{y}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{\frac{t}{y}}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-1}{\frac{t}{y}} - x}{\mathsf{neg}\left(\color{blue}{\left(x + 1\right)}\right)} \]

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(1 + x\right)}}{-1 - x} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \color{blue}{\left(1 + x\right)}}{-1 - x} \]

       2. lower-+.f64 52.8

          \[\leadsto \frac{-1 \cdot \left(1 + \color{blue}{x}\right)}{-1 - x} \]

   11. Applied rewrites 52.8%

       \[\leadsto \frac{\color{blue}{-1 \cdot \left(1 + x\right)}}{-1 - x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x - -1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{-1 \cdot \left(1 + x\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (- x -1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 5e-19)
     t_1
     (if (<= t_2 2.0) (/ (* -1.0 (+ 1.0 x)) (- -1.0 x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 5e-19) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((y / t) + x) / (x - (-1.0d0))
    t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_2 <= 5d-19) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = ((-1.0d0) * (1.0d0 + x)) / ((-1.0d0) - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x - -1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= 5e-19) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((y / t) + x) / (x - -1.0)
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= 5e-19:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= 5e-19)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(-1.0 * Float64(1.0 + x)) / Float64(-1.0 - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((y / t) + x) / (x - -1.0);
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= 5e-19)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-19], t$95$1, If[LessEqual[t$95$2, 2.0], N[(N[(-1.0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e-19 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 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.5

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.5%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x + 1} \]

      3. lower-+.f64 70.5

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x + 1} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + 1\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite=>}\left(add-flip, \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{x - \mathsf{Rewrite=>}\left(metadata-eval, -1\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{y}{t} + \color{blue}{x}}{\mathsf{Rewrite<=}\left(lift--.f64, \left(x - -1\right)\right)} \]

   8. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{t} + x}{x - -1}} \]

   if 5.0000000000000004e-19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. div-flip N/A

         \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      4. lower-unsound-/.f64 88.7

         \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{y \cdot z} - x}}}{x + 1} \]

      6. *-commutative N/A

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{z \cdot y} - x}}}{x + 1} \]

      7. lower-*.f64 88.7

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{z \cdot y} - x}}}{x + 1} \]

   3. Applied rewrites 88.7%

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{z \cdot y - x}}}}{x + 1} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t}{y}}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-/.f64 70.4

         \[\leadsto \frac{x + \frac{1}{\frac{t}{\color{blue}{y}}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t}{y}}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{1}{\frac{t}{y}}}{x + 1}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(x + \frac{1}{\frac{t}{y}}\right)\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(x + \frac{1}{\frac{t}{y}}\right)\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x + \frac{1}{\frac{t}{y}}\right)}\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      5. add-flip N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x - \left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      6. sub-negate N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right) - x}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right) - x}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{t}{y}}}\right)\right) - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      9. distribute-neg-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{t}{y}}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{-1}}{\frac{t}{y}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{\frac{t}{y}}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-1}{\frac{t}{y}} - x}{\mathsf{neg}\left(\color{blue}{\left(x + 1\right)}\right)} \]

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(1 + x\right)}}{-1 - x} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \color{blue}{\left(1 + x\right)}}{-1 - x} \]

       2. lower-+.f64 52.8

          \[\leadsto \frac{-1 \cdot \left(1 + \color{blue}{x}\right)}{-1 - x} \]

   11. Applied rewrites 52.8%

       \[\leadsto \frac{\color{blue}{-1 \cdot \left(1 + x\right)}}{-1 - x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{-1 \cdot \left(1 + x\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot \left(1 + x\right)}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 5e-19)
     (/ (+ x (/ y t)) 1.0)
     (if (<= t_1 2.0)
       (/ (* -1.0 (+ 1.0 x)) (- -1.0 x))
       (/ y (* t (+ 1.0 x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= 5e-19) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} 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 <= 5d-19) then
        tmp = (x + (y / t)) / 1.0d0
    else if (t_1 <= 2.0d0) then
        tmp = ((-1.0d0) * (1.0d0 + x)) / ((-1.0d0) - x)
    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 <= 5e-19) {
		tmp = (x + (y / t)) / 1.0;
	} else if (t_1 <= 2.0) {
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	} 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 <= 5e-19:
		tmp = (x + (y / t)) / 1.0
	elif t_1 <= 2.0:
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x)
	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 <= 5e-19)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(-1.0 * Float64(1.0 + x)) / Float64(-1.0 - x));
	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 <= 5e-19)
		tmp = (x + (y / t)) / 1.0;
	elseif (t_1 <= 2.0)
		tmp = (-1.0 * (1.0 + x)) / (-1.0 - x);
	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, 5e-19], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(-1.0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], 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))) < 5.0000000000000004e-19

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.5

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.5%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.0%

      \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{1}} \]

   if 5.0000000000000004e-19 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. div-flip N/A

         \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      4. lower-unsound-/.f64 88.7

         \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{y \cdot z} - x}}}{x + 1} \]

      6. *-commutative N/A

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{z \cdot y} - x}}}{x + 1} \]

      7. lower-*.f64 88.7

         \[\leadsto \frac{x + \frac{1}{\frac{t \cdot z - x}{\color{blue}{z \cdot y} - x}}}{x + 1} \]

   3. Applied rewrites 88.7%

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{z \cdot y - x}}}}{x + 1} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t}{y}}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-/.f64 70.4

         \[\leadsto \frac{x + \frac{1}{\frac{t}{\color{blue}{y}}}}{x + 1} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{t}{y}}}}{x + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + \frac{1}{\frac{t}{y}}}{x + 1}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(x + \frac{1}{\frac{t}{y}}\right)\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(x + \frac{1}{\frac{t}{y}}\right)\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x + \frac{1}{\frac{t}{y}}\right)}\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      5. add-flip N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x - \left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      6. sub-negate N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right) - x}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{t}{y}}\right)\right) - x}}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{t}{y}}}\right)\right) - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      9. distribute-neg-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{t}{y}}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{-1}}{\frac{t}{y}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{\frac{t}{y}}} - x}{\mathsf{neg}\left(\left(x + 1\right)\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-1}{\frac{t}{y}} - x}{\mathsf{neg}\left(\color{blue}{\left(x + 1\right)}\right)} \]

   8. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{t}{y}} - x}{-1 - x}} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(1 + x\right)}}{-1 - x} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \color{blue}{\left(1 + x\right)}}{-1 - x} \]

       2. lower-+.f64 52.8

          \[\leadsto \frac{-1 \cdot \left(1 + \color{blue}{x}\right)}{-1 - x} \]

   11. Applied rewrites 52.8%

       \[\leadsto \frac{\color{blue}{-1 \cdot \left(1 + x\right)}}{-1 - x} \]

   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 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 28.9

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y}{\color{blue}{t \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{t \cdot \color{blue}{\left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{t \cdot \left(1 + \color{blue}{x}\right)} \]

      3. lower-+.f64 27.4

         \[\leadsto \frac{y}{t \cdot \left(1 + x\right)} \]

   7. Applied rewrites 27.4%

      \[\leadsto \frac{y}{\color{blue}{t \cdot \left(1 + x\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 77.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{x - -1}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (- x -1.0))))
   (if (<= x -1.15e-13) t_1 (if (<= x 6.5e-9) (/ (+ x (/ y t)) 1.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x - -1.0);
	double tmp;
	if (x <= -1.15e-13) {
		tmp = t_1;
	} else if (x <= 6.5e-9) {
		tmp = (x + (y / t)) / 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) :: tmp
    t_1 = x / (x - (-1.0d0))
    if (x <= (-1.15d-13)) then
        tmp = t_1
    else if (x <= 6.5d-9) then
        tmp = (x + (y / t)) / 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (x - -1.0);
	double tmp;
	if (x <= -1.15e-13) {
		tmp = t_1;
	} else if (x <= 6.5e-9) {
		tmp = (x + (y / t)) / 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x - -1.0)
	tmp = 0
	if x <= -1.15e-13:
		tmp = t_1
	elif x <= 6.5e-9:
		tmp = (x + (y / t)) / 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x - -1.0))
	tmp = 0.0
	if (x <= -1.15e-13)
		tmp = t_1;
	elseif (x <= 6.5e-9)
		tmp = Float64(Float64(x + Float64(y / t)) / 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x - -1.0);
	tmp = 0.0;
	if (x <= -1.15e-13)
		tmp = t_1;
	elseif (x <= 6.5e-9)
		tmp = (x + (y / t)) / 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e-13], t$95$1, If[LessEqual[x, 6.5e-9], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999e-13 or 6.5000000000000003e-9 < x

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.1

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{x + \color{blue}{1}} \]

      3. add-flip N/A

         \[\leadsto \frac{x}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x}{x - -1} \]

      5. lift--.f64 55.1

         \[\leadsto \frac{x}{x - \color{blue}{-1}} \]

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x}{\color{blue}{x - -1}} \]

   if -1.1499999999999999e-13 < x < 6.5000000000000003e-9

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

      2. lower-/.f64 70.5

         \[\leadsto \frac{x + \frac{y}{\color{blue}{t}}}{x + 1} \]

   6. Applied rewrites 70.5%

      \[\leadsto \frac{\color{blue}{x + \frac{y}{t}}}{x + 1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.0%

      \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{y}{t \cdot \left(1 + x\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

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-8) t_1 (if (<= t_2 1.0) (/ x (- x -1.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (1.0 + x));
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e-8) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y / (t * (1.0d0 + x))
    t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_2 <= (-2d-8)) then
        tmp = t_1
    else if (t_2 <= 1.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (1.0 + x));
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -2e-8) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (t * (1.0 + x))
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= -2e-8:
		tmp = t_1
	elif t_2 <= 1.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(t * Float64(1.0 + x)))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -2e-8)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / (t * (1.0 + x));
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -2e-8)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(t * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-8], t$95$1, If[LessEqual[t$95$2, 1.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e-8 or 1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\left(1 + x\right)} \cdot \left(t \cdot z - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \color{blue}{\left(t \cdot z - x\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(\color{blue}{t \cdot z} - x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - \color{blue}{x}\right)} \]

      6. lower-*.f64 28.9

         \[\leadsto \frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)} \]

   4. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{y}{\color{blue}{t \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{t \cdot \color{blue}{\left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{t \cdot \left(1 + \color{blue}{x}\right)} \]

      3. lower-+.f64 27.4

         \[\leadsto \frac{y}{t \cdot \left(1 + x\right)} \]

   7. Applied rewrites 27.4%

      \[\leadsto \frac{y}{\color{blue}{t \cdot \left(1 + x\right)}} \]

   if -2e-8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.1

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{x + \color{blue}{1}} \]

      3. add-flip N/A

         \[\leadsto \frac{x}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x}{x - -1} \]

      5. lift--.f64 55.1

         \[\leadsto \frac{x}{x - \color{blue}{-1}} \]

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x}{\color{blue}{x - -1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]

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-8) (/ y t) (if (<= t_1 1.0) (/ x (- x -1.0)) (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e-8) {
		tmp = y / t;
	} else if (t_1 <= 1.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
    if (t_1 <= (-2d-8)) then
        tmp = y / t
    else if (t_1 <= 1.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -2e-8) {
		tmp = y / t;
	} else if (t_1 <= 1.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	tmp = 0
	if t_1 <= -2e-8:
		tmp = y / t
	elif t_1 <= 1.0:
		tmp = x / (x - -1.0)
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -2e-8)
		tmp = Float64(y / t);
	elseif (t_1 <= 1.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -2e-8)
		tmp = y / t;
	elseif (t_1 <= 1.0)
		tmp = x / (x - -1.0);
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-8], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(y / t), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -2e-8 or 1 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 25.4

         \[\leadsto \frac{y}{\color{blue}{t}} \]

   4. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{y}{t}} \]

   if -2e-8 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.1

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{x + \color{blue}{1}} \]

      3. add-flip N/A

         \[\leadsto \frac{x}{x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x}{x - -1} \]

      5. lift--.f64 55.1

         \[\leadsto \frac{x}{x - \color{blue}{-1}} \]

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x}{\color{blue}{x - -1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := 1 - \frac{1}{x}\\ \mathbf{if}\;x \leq -0.99:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ 1.0 x))))
   (if (<= x -0.99) t_1 (if (<= x 0.2) (/ y t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (1.0 / x);
	double tmp;
	if (x <= -0.99) {
		tmp = t_1;
	} else if (x <= 0.2) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - (1.0d0 / x)
    if (x <= (-0.99d0)) then
        tmp = t_1
    else if (x <= 0.2d0) then
        tmp = y / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (1.0 / x);
	double tmp;
	if (x <= -0.99) {
		tmp = t_1;
	} else if (x <= 0.2) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - (1.0 / x)
	tmp = 0
	if x <= -0.99:
		tmp = t_1
	elif x <= 0.2:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(1.0 / x))
	tmp = 0.0
	if (x <= -0.99)
		tmp = t_1;
	elseif (x <= 0.2)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - (1.0 / x);
	tmp = 0.0;
	if (x <= -0.99)
		tmp = t_1;
	elseif (x <= 0.2)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.99], t$95$1, If[LessEqual[x, 0.2], N[(y / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.98999999999999999 or 0.20000000000000001 < x

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + \frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z - x}{t \cdot z - x}}}{x + 1} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{x + \frac{\color{blue}{y \cdot z - x}}{t \cdot z - x}}{x + 1} \]

      4. div-sub N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1} \]

      5. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} + x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{y \cdot z}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{z \cdot y}}{t \cdot z - x} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot \frac{y}{t \cdot z - x}} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{y}{t \cdot z - x} \cdot z} + x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{y}{t \cdot z - x}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1} \]

      14. lower-/.f64 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\frac{x}{t \cdot z - x}}}{x + 1} \]

   3. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}{x + 1} \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

      2. lower-+.f64 55.1

         \[\leadsto \frac{x}{1 + \color{blue}{x}} \]

   6. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

   7. Taylor expanded in x around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \frac{1}{\color{blue}{x}} \]

      2. lower-/.f64 45.0

         \[\leadsto 1 - \frac{1}{x} \]

   9. Applied rewrites 45.0%

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]

   if -0.98999999999999999 < x < 0.20000000000000001

   1. Initial program 88.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 25.4

         \[\leadsto \frac{y}{\color{blue}{t}} \]

   4. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 25.4% accurate, 5.6× speedup

Math

\[\frac{y}{t} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ y t))

C

double code(double x, double y, double z, double t) {
	return y / t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y / t
end function

Java

public static double code(double x, double y, double z, double t) {
	return y / t;
}

Python

def code(x, y, z, t):
	return y / t

Julia

function code(x, y, z, t)
	return Float64(y / t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y / t;
end

Wolfram

code[x_, y_, z_, t_] := N[(y / t), $MachinePrecision]

Derivation

1. Initial program 88.8%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Step-by-step derivation

   1. lower-/.f64 25.4

      \[\leadsto \frac{y}{\color{blue}{t}} \]

4. Applied rewrites 25.4%

   \[\leadsto \color{blue}{\frac{y}{t}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025167 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))