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

Percentage Accurate: 89.3% → 98.7%

Time: 7.6s

Alternatives: 14

Speedup: 0.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + y \cdot \frac{z}{t\_1}}{x - -1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x - -1}\\ \mathbf{if}\;t\_3 \leq -5:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (+ x (* y (/ z t_1))) (- x -1.0)))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (- x -1.0))))
   (if (<= t_3 -5.0)
     t_2
     (if (<= t_3 4e-7)
       (/ (- x (/ (- (/ x z) y) t)) (- x -1.0))
       (if (<= t_3 2.0)
         (/ (- x (/ x (fma t z (- x)))) (- x -1.0))
         (if (<= t_3 INFINITY) t_2 (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x - -1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_3 <= -5.0) {
		tmp = t_2;
	} else if (t_3 <= 4e-7) {
		tmp = (x - (((x / z) - y) / t)) / (x - -1.0);
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (x - -1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		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(y * Float64(z / t_1))) / Float64(x - -1.0))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_3 <= -5.0)
		tmp = t_2;
	elseif (t_3 <= 4e-7)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x - -1.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(x - -1.0));
	elseif (t_3 <= Inf)
		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[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5.0], t$95$2, If[LessEqual[t$95$3, 4e-7], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5 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 79.9%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. *-lft-identity N/A

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

      20. metadata-eval N/A

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

      21. fp-cancel-sign-sub-inv N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.6%

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

   if -5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e-7

   1. Initial program 95.0%

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

   3. Taylor expanded in t around -inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 3.9999999999999998e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -5:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := t \cdot z - x\\ t_3 := \frac{x + y \cdot \frac{z}{t\_2}}{x - -1}\\ t_4 := \frac{x + \frac{t\_1}{t\_2}}{x - -1}\\ \mathbf{if}\;t\_4 \leq -5:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{t\_1}{t \cdot z}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* t z) x))
        (t_3 (/ (+ x (* y (/ z t_2))) (- x -1.0)))
        (t_4 (/ (+ x (/ t_1 t_2)) (- x -1.0))))
   (if (<= t_4 -5.0)
     t_3
     (if (<= t_4 4e-7)
       (/ (+ x (/ t_1 (* t z))) (- x -1.0))
       (if (<= t_4 2.0)
         (/ (- x (/ x (fma t z (- x)))) (- x -1.0))
         (if (<= t_4 INFINITY) t_3 (/ (+ (/ y t) x) (- x -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (t * z) - x;
	double t_3 = (x + (y * (z / t_2))) / (x - -1.0);
	double t_4 = (x + (t_1 / t_2)) / (x - -1.0);
	double tmp;
	if (t_4 <= -5.0) {
		tmp = t_3;
	} else if (t_4 <= 4e-7) {
		tmp = (x + (t_1 / (t * z))) / (x - -1.0);
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (x - -1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - x)
	t_2 = Float64(Float64(t * z) - x)
	t_3 = Float64(Float64(x + Float64(y * Float64(z / t_2))) / Float64(x - -1.0))
	t_4 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_4 <= -5.0)
		tmp = t_3;
	elseif (t_4 <= 4e-7)
		tmp = Float64(Float64(x + Float64(t_1 / Float64(t * z))) / Float64(x - -1.0));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(x - -1.0));
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(y * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5.0], t$95$3, If[LessEqual[t$95$4, 4e-7], N[(N[(x + N[(t$95$1 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5 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 79.9%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. *-lft-identity N/A

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

      20. metadata-eval N/A

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

      21. fp-cancel-sign-sub-inv N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.6%

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

   if -5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999998e-7

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if 3.9999999999999998e-7 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -5:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + y \cdot \frac{z}{t\_1}}{x - -1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x - -1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+68}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{y}{1 + x}, -1, \frac{x}{\mathsf{fma}\left(z, x, z\right)}\right)}{t}, -1, \frac{x}{1 + x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (+ x (* y (/ z t_1))) (- x -1.0)))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (- x -1.0))))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 2e+68)
       t_3
       (if (<= t_3 INFINITY)
         t_2
         (fma
          (/ (fma (/ y (+ 1.0 x)) -1.0 (/ x (fma z x z))) t)
          -1.0
          (/ x (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x - -1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= 2e+68) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma((fma((y / (1.0 + x)), -1.0, (x / fma(z, x, z))) / t), -1.0, (x / (1.0 + x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(y * Float64(z / t_1))) / Float64(x - -1.0))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= 2e+68)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = fma(Float64(fma(Float64(y / Float64(1.0 + x)), -1.0, Float64(x / fma(z, x, z))) / t), -1.0, Float64(x / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, 2e+68], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$2, N[(N[(N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(x / N[(z * x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * -1.0 + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 1.99999999999999991e68 < (/.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 60.6%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. *-lft-identity N/A

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

      20. metadata-eval N/A

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

      21. fp-cancel-sign-sub-inv N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.8%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e68

   1. Initial program 99.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{y}{1 + x}, -1, \frac{x}{\mathsf{fma}\left(z, x, z\right)}\right)}{t}, -1, \frac{x}{1 + x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + y \cdot \frac{z}{t\_1}}{x - -1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x - -1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+68}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (+ x (* y (/ z t_1))) (- x -1.0)))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (- x -1.0))))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 2e+68)
       t_3
       (if (<= t_3 INFINITY) t_2 (/ (+ (/ y t) x) (- x -1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x - -1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= 2e+68) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x - -1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_3 <= 2e+68) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * z) - x
	t_2 = (x + (y * (z / t_1))) / (x - -1.0)
	t_3 = (x + (((y * z) - x) / t_1)) / (x - -1.0)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_2
	elif t_3 <= 2e+68:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = ((y / t) + x) / (x - -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * z) - x)
	t_2 = Float64(Float64(x + Float64(y * Float64(z / t_1))) / Float64(x - -1.0))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= 2e+68)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * z) - x;
	t_2 = (x + (y * (z / t_1))) / (x - -1.0);
	t_3 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_2;
	elseif (t_3 <= 2e+68)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = ((y / t) + x) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, 2e+68], t$95$3, If[LessEqual[t$95$3, Infinity], 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))) < -inf.0 or 1.99999999999999991e68 < (/.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 60.6%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. *-lft-identity N/A

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

      20. metadata-eval N/A

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

      21. fp-cancel-sign-sub-inv N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.8%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e68

   1. Initial program 99.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -\infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot z - x\\ t_2 := \frac{x + y \cdot \frac{z}{t\_1}}{x - -1}\\ t_3 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x - -1}\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z) x))
        (t_2 (/ (+ x (* y (/ z t_1))) (- x -1.0)))
        (t_3 (/ (+ x (/ (- (* y z) x) t_1)) (- x -1.0))))
   (if (<= t_3 2e-27)
     t_2
     (if (<= t_3 2.0)
       (/ (- x (/ x (fma t z (- x)))) (- x -1.0))
       (if (<= t_3 INFINITY) t_2 (/ (+ (/ y t) x) (- x -1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * z) - x;
	double t_2 = (x + (y * (z / t_1))) / (x - -1.0);
	double t_3 = (x + (((y * z) - x) / t_1)) / (x - -1.0);
	double tmp;
	if (t_3 <= 2e-27) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = (x - (x / fma(t, z, -x))) / (x - -1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		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(y * Float64(z / t_1))) / Float64(x - -1.0))
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_3 <= 2e-27)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(x - -1.0));
	elseif (t_3 <= Inf)
		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[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-27], t$95$2, If[LessEqual[t$95$3, 2.0], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], 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))) < 2.0000000000000001e-27 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 85.1%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. *-lft-identity N/A

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

      20. metadata-eval N/A

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

      21. fp-cancel-sign-sub-inv N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 89.5%

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

   if 2.0000000000000001e-27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq \infty:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0)))
        (t_2 (fma t z (- x)))
        (t_3 (/ (* y (/ z t_2)) (- x -1.0))))
   (if (<= t_1 -5.0)
     t_3
     (if (<= t_1 2.0)
       (/ (- x (/ x t_2)) (- x -1.0))
       (if (<= t_1 INFINITY) t_3 (/ (+ (/ 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 t_2 = fma(t, z, -x);
	double t_3 = (y * (z / t_2)) / (x - -1.0);
	double tmp;
	if (t_1 <= -5.0) {
		tmp = t_3;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} 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))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(y * Float64(z / t_2)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -5.0)
		tmp = t_3;
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x - -1.0));
	elseif (t_1 <= Inf)
		tmp = t_3;
	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[(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[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5.0], t$95$3, If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, 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 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 79.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 92.1

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

   5. Applied rewrites 92.1%

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

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

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 92.9

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

   5. Applied rewrites 92.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.0%

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

Alternative 7: 88.4% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0)))
        (t_2 (fma t z (- x)))
        (t_3 (* (/ y t_2) (/ z (+ 1.0 x)))))
   (if (<= t_1 -5.0)
     t_3
     (if (<= t_1 2.0)
       (/ (- x (/ x t_2)) (- x -1.0))
       (if (<= t_1 INFINITY) t_3 (/ (+ (/ 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 t_2 = fma(t, z, -x);
	double t_3 = (y / t_2) * (z / (1.0 + x));
	double tmp;
	if (t_1 <= -5.0) {
		tmp = t_3;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x - -1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} 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))
	t_2 = fma(t, z, Float64(-x))
	t_3 = Float64(Float64(y / t_2) * Float64(z / Float64(1.0 + x)))
	tmp = 0.0
	if (t_1 <= -5.0)
		tmp = t_3;
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x - -1.0));
	elseif (t_1 <= Inf)
		tmp = t_3;
	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[(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[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / t$95$2), $MachinePrecision] * N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5.0], t$95$3, If[LessEqual[t$95$1, 2.0], N[(N[(x - N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, 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 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 79.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 79.5

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

   5. Applied rewrites 79.5%

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

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

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 92.9

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

   5. Applied rewrites 92.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 89.7%

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

Alternative 8: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999942764:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 -4e+36)
     (/ y t)
     (if (<= t_1 0.9999999999942764)
       (/ x (+ 1.0 x))
       (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -4e+36) {
		tmp = y / t;
	} else if (t_1 <= 0.9999999999942764) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - (-1.0d0))
    if (t_1 <= (-4d+36)) then
        tmp = y / t
    else if (t_1 <= 0.9999999999942764d0) then
        tmp = x / (1.0d0 + x)
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -4e+36) {
		tmp = y / t;
	} else if (t_1 <= 0.9999999999942764) {
		tmp = x / (1.0 + x);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if t_1 <= -4e+36:
		tmp = y / t
	elif t_1 <= 0.9999999999942764:
		tmp = x / (1.0 + x)
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -4e+36)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.9999999999942764)
		tmp = Float64(x / Float64(1.0 + x));
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if (t_1 <= -4e+36)
		tmp = y / t;
	elseif (t_1 <= 0.9999999999942764)
		tmp = x / (1.0 + x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+36], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999942764], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000017e36 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 66.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 51.5

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

   5. Applied rewrites 51.5%

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

   if -4.00000000000000017e36 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999999999427636

   1. Initial program 95.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 49.2

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

   5. Applied rewrites 49.2%

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

   if 0.99999999999427636 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -4 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 0.9999999999942764:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 -4e+36)
     (/ y t)
     (if (<= t_1 2e-6)
       (* (fma (- x 1.0) x 1.0) x)
       (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -4e+36) {
		tmp = y / t;
	} else if (t_1 <= 2e-6) {
		tmp = fma((x - 1.0), x, 1.0) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -4e+36)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-6)
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return 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+36], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-6], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000017e36 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 66.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 51.5

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

   5. Applied rewrites 51.5%

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

   if -4.00000000000000017e36 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e-6

   1. Initial program 95.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 46.8

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.9%

      \[\leadsto \mathsf{fma}\left(x - 1, x, 1\right) \cdot \color{blue}{x} \]

   if 1.99999999999999991e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -4 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (<= t_1 -4e+36)
     (/ y t)
     (if (<= t_1 2e-6) (* (fma -1.0 x 1.0) x) (if (<= t_1 2.0) 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if (t_1 <= -4e+36) {
		tmp = y / t;
	} else if (t_1 <= 2e-6) {
		tmp = fma(-1.0, x, 1.0) * x;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = y / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if (t_1 <= -4e+36)
		tmp = Float64(y / t);
	elseif (t_1 <= 2e-6)
		tmp = Float64(fma(-1.0, x, 1.0) * x);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(y / t);
	end
	return 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+36], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 2e-6], N[(N[(-1.0 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, N[(y / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000017e36 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 66.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 51.5

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

   5. Applied rewrites 51.5%

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

   if -4.00000000000000017e36 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e-6

   1. Initial program 95.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 46.8

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.4%

      \[\leadsto \mathsf{fma}\left(-1, x, 1\right) \cdot \color{blue}{x} \]

   if 1.99999999999999991e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq -4 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-1, x, 1\right) \cdot x\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-27} \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (or (<= t_1 2e-27) (not (<= t_1 2.0)))
     (/ (+ (/ y t) x) (- x -1.0))
     (/ (- x (/ x (fma t z (- 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 <= 2e-27) || !(t_1 <= 2.0)) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else {
		tmp = (x - (x / fma(t, z, -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 <= 2e-27) || !(t_1 <= 2.0))
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	else
		tmp = Float64(Float64(x - Float64(x / fma(t, z, Float64(-x)))) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 2e-27], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(x / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e-27 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if 2.0000000000000001e-27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-27} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1}\\ \mathbf{if}\;t\_1 \leq 0.9999999999942764 \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0))))
   (if (or (<= t_1 0.9999999999942764) (not (<= t_1 2.0)))
     (/ (+ (/ y t) x) (- x -1.0))
     1.0)))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if ((t_1 <= 0.9999999999942764) || !(t_1 <= 2.0)) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - (-1.0d0))
    if ((t_1 <= 0.9999999999942764d0) .or. (.not. (t_1 <= 2.0d0))) then
        tmp = ((y / t) + x) / (x - (-1.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	double tmp;
	if ((t_1 <= 0.9999999999942764) || !(t_1 <= 2.0)) {
		tmp = ((y / t) + x) / (x - -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0)
	tmp = 0
	if (t_1 <= 0.9999999999942764) or not (t_1 <= 2.0):
		tmp = ((y / t) + x) / (x - -1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x - -1.0))
	tmp = 0.0
	if ((t_1 <= 0.9999999999942764) || !(t_1 <= 2.0))
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x - -1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0);
	tmp = 0.0;
	if ((t_1 <= 0.9999999999942764) || ~((t_1 <= 2.0)))
		tmp = ((y / t) + x) / (x - -1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.9999999999942764], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.99999999999427636 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 77.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if 0.99999999999427636 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 0.9999999999942764 \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-1, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (- x -1.0)) 2e-6)
   (* (fma -1.0 x 1.0) x)
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((t * z) - x))) / (x - -1.0)) <= 2e-6) {
		tmp = fma(-1.0, x, 1.0) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x - -1.0)) <= 2e-6)
		tmp = Float64(fma(-1.0, x, 1.0) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[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], 2e-6], N[(N[(-1.0 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e-6

   1. Initial program 90.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 26.0%

      \[\leadsto \mathsf{fma}\left(-1, x, 1\right) \cdot \color{blue}{x} \]

   if 1.99999999999999991e-6 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 80.1%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x - -1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-1, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.1% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 56.7%

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

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025008 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

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

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