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

Percentage Accurate: 89.0% → 96.1%

Time: 11.8s

Alternatives: 20

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{x}{t}, -1, z\right) \cdot t} \cdot \frac{z}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (* (/ y (* (fma (/ x t) -1.0 z) t)) (/ z (+ 1.0 x)))
     (if (<= t_1 2e+271) t_1 (/ (+ (/ y t) x) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / (fma((x / t), -1.0, z) * t)) * (z / (1.0 + x));
	} else if (t_1 <= 2e+271) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / Float64(fma(Float64(x / t), -1.0, z) * t)) * Float64(z / Float64(1.0 + x)));
	elseif (t_1 <= 2e+271)
		tmp = t_1;
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(N[(N[(x / t), $MachinePrecision] * -1.0 + z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+271], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 38.2%

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

   3. Taylor expanded in y around inf

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

      1. *-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 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 78.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e271

   1. Initial program 99.4%

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

   if 1.99999999999999991e271 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 32.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 93.0

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

   5. Applied rewrites 93.0%

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

Alternative 2: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{y \cdot \frac{z}{t\_2}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{x - \frac{\frac{x}{z} - y}{t}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

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 -1.5e+30)
     t_3
     (if (<= t_1 0.05)
       (/ (- x (/ (- (/ x z) y) t)) (+ x 1.0))
       (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 <= -1.5e+30) {
		tmp = t_3;
	} else if (t_1 <= 0.05) {
		tmp = (x - (((x / z) - y) / t)) / (x + 1.0);
	} 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 <= -1.5e+30)
		tmp = t_3;
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(x - Float64(Float64(Float64(x / z) - y) / t)) / Float64(x + 1.0));
	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, -1.5e+30], t$95$3, If[LessEqual[t$95$1, 0.05], N[(N[(x - N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 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 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.49999999999999989e30 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 78.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

   if -1.49999999999999989e30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   1. Initial program 97.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 0.050000000000000003 < (/.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 98.4

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

   5. Applied rewrites 98.4%

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

   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 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 3: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, -x\right)\\ t_2 := \frac{y \cdot \frac{z}{t\_1}}{x + 1}\\ t_3 := y \cdot z - x\\ t_4 := \frac{x + \frac{t\_3}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{x + \frac{t\_3}{t \cdot z}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma t z (- x)))
        (t_2 (/ (* y (/ z t_1)) (+ x 1.0)))
        (t_3 (- (* y z) x))
        (t_4 (/ (+ x (/ t_3 (- (* t z) x))) (+ x 1.0))))
   (if (<= t_4 -1.5e+30)
     t_2
     (if (<= t_4 0.05)
       (/ (+ x (/ t_3 (* t z))) (+ x 1.0))
       (if (<= t_4 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_2 (/ (+ (/ y t) x) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, z, -x);
	double t_2 = (y * (z / t_1)) / (x + 1.0);
	double t_3 = (y * z) - x;
	double t_4 = (x + (t_3 / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_4 <= -1.5e+30) {
		tmp = t_2;
	} else if (t_4 <= 0.05) {
		tmp = (x + (t_3 / (t * z))) / (x + 1.0);
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, z, Float64(-x))
	t_2 = Float64(Float64(y * Float64(z / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(y * z) - x)
	t_4 = Float64(Float64(x + Float64(t_3 / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -1.5e+30)
		tmp = t_2;
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(x + Float64(t_3 / Float64(t * z))) / Float64(x + 1.0));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(t$95$3 / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.5e+30], t$95$2, If[LessEqual[t$95$4, 0.05], N[(N[(x + N[(t$95$3 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.49999999999999989e30 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 78.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

   if -1.49999999999999989e30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.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 98.4

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

   5. Applied rewrites 98.4%

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

   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 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{y \cdot \frac{z}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 0.05:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{y \cdot \frac{z}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, -x\right)\\ t_2 := \frac{y \cdot \frac{z}{t\_1}}{x + 1}\\ t_3 := y \cdot z - x\\ t_4 := \frac{x + \frac{t\_3}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_4 \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{x + \frac{t\_3}{t \cdot z}}{1}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_1}}{x + 1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma t z (- x)))
        (t_2 (/ (* y (/ z t_1)) (+ x 1.0)))
        (t_3 (- (* y z) x))
        (t_4 (/ (+ x (/ t_3 (- (* t z) x))) (+ x 1.0))))
   (if (<= t_4 -1.5e+30)
     t_2
     (if (<= t_4 0.05)
       (/ (+ x (/ t_3 (* t z))) 1.0)
       (if (<= t_4 2.0)
         (/ (- x (/ x t_1)) (+ x 1.0))
         (if (<= t_4 INFINITY) t_2 (/ (+ (/ y t) x) (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, z, -x);
	double t_2 = (y * (z / t_1)) / (x + 1.0);
	double t_3 = (y * z) - x;
	double t_4 = (x + (t_3 / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_4 <= -1.5e+30) {
		tmp = t_2;
	} else if (t_4 <= 0.05) {
		tmp = (x + (t_3 / (t * z))) / 1.0;
	} else if (t_4 <= 2.0) {
		tmp = (x - (x / t_1)) / (x + 1.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, z, Float64(-x))
	t_2 = Float64(Float64(y * Float64(z / t_1)) / Float64(x + 1.0))
	t_3 = Float64(Float64(y * z) - x)
	t_4 = Float64(Float64(x + Float64(t_3 / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_4 <= -1.5e+30)
		tmp = t_2;
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(x + Float64(t_3 / Float64(t * z))) / 1.0);
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * z + (-x)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + N[(t$95$3 / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.5e+30], t$95$2, If[LessEqual[t$95$4, 0.05], N[(N[(x + N[(t$95$3 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.49999999999999989e30 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 78.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

   if -1.49999999999999989e30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.8%

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

   if 0.050000000000000003 < (/.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 98.4

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

   5. Applied rewrites 98.4%

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

   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 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{y \cdot \frac{z}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 0.05:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z}}{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq \infty:\\ \;\;\;\;\frac{y \cdot \frac{z}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{y \cdot \frac{z}{t\_2}}{x + 1}\\ t_4 := \frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-117}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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)))
        (t_4 (/ (+ (/ y t) x) (+ x 1.0))))
   (if (<= t_1 -1.5e+30)
     t_3
     (if (<= t_1 5e-117)
       t_4
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_1 INFINITY) t_3 t_4))))))

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 t_4 = ((y / t) + x) / (x + 1.0);
	double tmp;
	if (t_1 <= -1.5e+30) {
		tmp = t_3;
	} else if (t_1 <= 5e-117) {
		tmp = t_4;
	} 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 = t_4;
	}
	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))
	t_4 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1.5e+30)
		tmp = t_3;
	elseif (t_1 <= 5e-117)
		tmp = t_4;
	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 = t_4;
	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]}, Block[{t$95$4 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.5e+30], t$95$3, If[LessEqual[t$95$1, 5e-117], t$95$4, 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, t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.49999999999999989e30 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 78.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

   if -1.49999999999999989e30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e-117 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 5e-117 < (/.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 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 6: 91.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{t} + x}{x + 1}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_3 := \mathsf{fma}\left(t, z, -x\right)\\ \mathbf{if}\;t\_2 \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{t\_3} \cdot \frac{z}{1 + x}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_3}}{x + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{z \cdot y}{t\_3 \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ y t) x) (+ x 1.0)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_3 (fma t z (- x))))
   (if (<= t_2 -1.5e+30)
     (* (/ y t_3) (/ z (+ 1.0 x)))
     (if (<= t_2 5e-117)
       t_1
       (if (<= t_2 2.0)
         (/ (- x (/ x t_3)) (+ x 1.0))
         (if (<= t_2 2e+271) (/ (* z y) (* t_3 (+ 1.0 x))) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((y / t) + x) / (x + 1.0);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_3 = fma(t, z, -x);
	double tmp;
	if (t_2 <= -1.5e+30) {
		tmp = (y / t_3) * (z / (1.0 + x));
	} else if (t_2 <= 5e-117) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = (x - (x / t_3)) / (x + 1.0);
	} else if (t_2 <= 2e+271) {
		tmp = (z * y) / (t_3 * (1.0 + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_3 = fma(t, z, Float64(-x))
	tmp = 0.0
	if (t_2 <= -1.5e+30)
		tmp = Float64(Float64(y / t_3) * Float64(z / Float64(1.0 + x)));
	elseif (t_2 <= 5e-117)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_3)) / Float64(x + 1.0));
	elseif (t_2 <= 2e+271)
		tmp = Float64(Float64(z * y) / Float64(t_3 * Float64(1.0 + x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * z + (-x)), $MachinePrecision]}, If[LessEqual[t$95$2, -1.5e+30], N[(N[(y / t$95$3), $MachinePrecision] * N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-117], t$95$1, If[LessEqual[t$95$2, 2.0], N[(N[(x - N[(x / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+271], N[(N[(z * y), $MachinePrecision] / N[(t$95$3 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.49999999999999989e30

   1. Initial program 74.2%

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

   3. Taylor expanded in y around inf

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

      1. *-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 80.0

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

   5. Applied rewrites 80.0%

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

   if -1.49999999999999989e30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e-117 or 1.99999999999999991e271 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   if 5e-117 < (/.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 97.2

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

   5. Applied rewrites 97.2%

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

   if 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e271

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 98.8%

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

Alternative 7: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_2 := \mathsf{fma}\left(t, z, -x\right)\\ t_3 := \frac{z \cdot y}{t\_2 \cdot \left(1 + x\right)}\\ t_4 := \frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-117}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x - \frac{x}{t\_2}}{x + 1}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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 (/ (* z y) (* t_2 (+ 1.0 x))))
        (t_4 (/ (+ (/ y t) x) (+ x 1.0))))
   (if (<= t_1 -1.5e+30)
     t_3
     (if (<= t_1 5e-117)
       t_4
       (if (<= t_1 2.0)
         (/ (- x (/ x t_2)) (+ x 1.0))
         (if (<= t_1 2e+271) t_3 t_4))))))

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 = (z * y) / (t_2 * (1.0 + x));
	double t_4 = ((y / t) + x) / (x + 1.0);
	double tmp;
	if (t_1 <= -1.5e+30) {
		tmp = t_3;
	} else if (t_1 <= 5e-117) {
		tmp = t_4;
	} else if (t_1 <= 2.0) {
		tmp = (x - (x / t_2)) / (x + 1.0);
	} else if (t_1 <= 2e+271) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	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(z * y) / Float64(t_2 * Float64(1.0 + x)))
	t_4 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1.5e+30)
		tmp = t_3;
	elseif (t_1 <= 5e-117)
		tmp = t_4;
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(x - Float64(x / t_2)) / Float64(x + 1.0));
	elseif (t_1 <= 2e+271)
		tmp = t_3;
	else
		tmp = t_4;
	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[(z * y), $MachinePrecision] / N[(t$95$2 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.5e+30], t$95$3, If[LessEqual[t$95$1, 5e-117], t$95$4, 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, 2e+271], t$95$3, t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.49999999999999989e30 or 2 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e271

   1. Initial program 84.7%

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

   3. Taylor expanded in y around inf

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

      1. *-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 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.3%

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

   if -1.49999999999999989e30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5e-117 or 1.99999999999999991e271 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 66.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   if 5e-117 < (/.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 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 8: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{\mathsf{fma}\left(t, z, -x\right) \cdot \left(1 + x\right)}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_3 := \frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1.5 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 200000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(x, x, x\right)}, -y, 1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z y) (* (fma t z (- x)) (+ 1.0 x))))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_3 (/ (+ (/ y t) x) (+ x 1.0))))
   (if (<= t_2 -1.5e+30)
     t_1
     (if (<= t_2 0.05)
       t_3
       (if (<= t_2 200000000000.0)
         (fma (/ z (fma x x x)) (- y) 1.0)
         (if (<= t_2 2e+271) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / (fma(t, z, -x) * (1.0 + x));
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_3 = ((y / t) + x) / (x + 1.0);
	double tmp;
	if (t_2 <= -1.5e+30) {
		tmp = t_1;
	} else if (t_2 <= 0.05) {
		tmp = t_3;
	} else if (t_2 <= 200000000000.0) {
		tmp = fma((z / fma(x, x, x)), -y, 1.0);
	} else if (t_2 <= 2e+271) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) / Float64(fma(t, z, Float64(-x)) * Float64(1.0 + x)))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_3 = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -1.5e+30)
		tmp = t_1;
	elseif (t_2 <= 0.05)
		tmp = t_3;
	elseif (t_2 <= 200000000000.0)
		tmp = fma(Float64(z / fma(x, x, x)), Float64(-y), 1.0);
	elseif (t_2 <= 2e+271)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / N[(N[(t * z + (-x)), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.5e+30], t$95$1, If[LessEqual[t$95$2, 0.05], t$95$3, If[LessEqual[t$95$2, 200000000000.0], N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * (-y) + 1.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+271], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -1.49999999999999989e30 or 2e11 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e271

   1. Initial program 84.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 84.7

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 84.4%

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

   if -1.49999999999999989e30 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003 or 1.99999999999999991e271 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   if 0.050000000000000003 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e11

   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 t around 0

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

      1. associate-+r+ N/A

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

      2. div-add N/A

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

      3. *-inverses N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

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

Alternative 9: 80.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\frac{\frac{y}{t} + x}{1}\\ \mathbf{elif}\;t\_2 \leq 200000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(x, x, x\right)}, -y, 1\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (+ 1.0 x) t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 -5e+99)
     t_1
     (if (<= t_2 0.05)
       (/ (+ (/ y t) x) 1.0)
       (if (<= t_2 200000000000.0)
         (fma (/ z (fma x x x)) (- y) 1.0)
         (if (<= t_2 INFINITY) t_1 (/ x (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+99) {
		tmp = t_1;
	} else if (t_2 <= 0.05) {
		tmp = ((y / t) + x) / 1.0;
	} else if (t_2 <= 200000000000.0) {
		tmp = fma((z / fma(x, x, x)), -y, 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(1.0 + x) * t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e+99)
		tmp = t_1;
	elseif (t_2 <= 0.05)
		tmp = Float64(Float64(Float64(y / t) + x) / 1.0);
	elseif (t_2 <= 200000000000.0)
		tmp = fma(Float64(z / fma(x, x, x)), Float64(-y), 1.0);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+99], t$95$1, If[LessEqual[t$95$2, 0.05], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$2, 200000000000.0], N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * (-y) + 1.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000008e99 or 2e11 < (/.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 76.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 71.1%

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

   if -5.00000000000000008e99 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   1. Initial program 97.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.5%

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

   if 0.050000000000000003 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e11

   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 t around 0

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

      1. associate-+r+ N/A

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

      2. div-add N/A

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

      3. *-inverses N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

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

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

   1. Initial program 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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 10: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot x}{\mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{elif}\;t\_2 \leq 200000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(x, x, x\right)}, -y, 1\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (+ 1.0 x) t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_2 -500000.0)
     t_1
     (if (<= t_2 0.05)
       (/ (* (- x 1.0) x) (fma x x -1.0))
       (if (<= t_2 200000000000.0)
         (fma (/ z (fma x x x)) (- y) 1.0)
         (if (<= t_2 INFINITY) t_1 (/ x (+ 1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.05) {
		tmp = ((x - 1.0) * x) / fma(x, x, -1.0);
	} else if (t_2 <= 200000000000.0) {
		tmp = fma((z / fma(x, x, x)), -y, 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(1.0 + x) * t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -500000.0)
		tmp = t_1;
	elseif (t_2 <= 0.05)
		tmp = Float64(Float64(Float64(x - 1.0) * x) / fma(x, x, -1.0));
	elseif (t_2 <= 200000000000.0)
		tmp = fma(Float64(z / fma(x, x, x)), Float64(-y), 1.0);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000.0], t$95$1, If[LessEqual[t$95$2, 0.05], N[(N[(N[(x - 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 200000000000.0], N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * (-y) + 1.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -5e5 or 2e11 < (/.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 78.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.3%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003

   1. Initial program 97.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. difference-of-squares-rev N/A

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

      16. difference-of-sqr--1-rev N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{\color{blue}{x \cdot \left(x - 1\right)}}{\mathsf{fma}\left(x, x, -1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 59.2

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

   7. Applied rewrites 59.2%

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

   if 0.050000000000000003 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e11

   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 t around 0

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

      1. associate-+r+ N/A

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

      2. div-add N/A

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

      3. *-inverses N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

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

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

   1. Initial program 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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 11: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_3 := \frac{x}{1 + x}\\ \mathbf{if}\;t\_2 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 200000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(x, x, x\right)}, -y, 1\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (+ 1.0 x) t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_3 (/ x (+ 1.0 x))))
   (if (<= t_2 -500000.0)
     t_1
     (if (<= t_2 0.05)
       t_3
       (if (<= t_2 200000000000.0)
         (fma (/ z (fma x x x)) (- y) 1.0)
         (if (<= t_2 INFINITY) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_3 = x / (1.0 + x);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.05) {
		tmp = t_3;
	} else if (t_2 <= 200000000000.0) {
		tmp = fma((z / fma(x, x, x)), -y, 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(1.0 + x) * t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_3 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= -500000.0)
		tmp = t_1;
	elseif (t_2 <= 0.05)
		tmp = t_3;
	elseif (t_2 <= 200000000000.0)
		tmp = fma(Float64(z / fma(x, x, x)), Float64(-y), 1.0);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000.0], t$95$1, If[LessEqual[t$95$2, 0.05], t$95$3, If[LessEqual[t$95$2, 200000000000.0], N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * (-y) + 1.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]]

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))) < -5e5 or 2e11 < (/.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 78.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.3%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 0.050000000000000003 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e11

   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 t around 0

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

      1. associate-+r+ N/A

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

      2. div-add N/A

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

      3. *-inverses N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

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

Alternative 12: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_3 := \frac{x}{1 + x}\\ \mathbf{if}\;t\_2 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 200000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{\mathsf{fma}\left(x, x, x\right)}, z, 1\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (+ 1.0 x) t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_3 (/ x (+ 1.0 x))))
   (if (<= t_2 -500000.0)
     t_1
     (if (<= t_2 0.05)
       t_3
       (if (<= t_2 200000000000.0)
         (fma (/ (- y) (fma x x x)) z 1.0)
         (if (<= t_2 INFINITY) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_3 = x / (1.0 + x);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.05) {
		tmp = t_3;
	} else if (t_2 <= 200000000000.0) {
		tmp = fma((-y / fma(x, x, x)), z, 1.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(1.0 + x) * t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_3 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= -500000.0)
		tmp = t_1;
	elseif (t_2 <= 0.05)
		tmp = t_3;
	elseif (t_2 <= 200000000000.0)
		tmp = fma(Float64(Float64(-y) / fma(x, x, x)), z, 1.0);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000.0], t$95$1, If[LessEqual[t$95$2, 0.05], t$95$3, If[LessEqual[t$95$2, 200000000000.0], N[(N[((-y) / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * z + 1.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]]

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))) < -5e5 or 2e11 < (/.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 78.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.3%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.050000000000000003 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 0.050000000000000003 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e11

   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 z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{y}{x + {x}^{2}}, z, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.3%

      \[\leadsto \mathsf{fma}\left(\frac{-y}{\mathsf{fma}\left(x, x, x\right)}, z, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\left(1 + x\right) \cdot t}\\ t_2 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_3 := \frac{x}{1 + x}\\ \mathbf{if}\;t\_2 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 200000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* (+ 1.0 x) t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_3 (/ x (+ 1.0 x))))
   (if (<= t_2 -500000.0)
     t_1
     (if (<= t_2 0.02)
       t_3
       (if (<= t_2 200000000000.0) 1.0 (if (<= t_2 INFINITY) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_3 = x / (1.0 + x);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.02) {
		tmp = t_3;
	} else if (t_2 <= 200000000000.0) {
		tmp = 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / ((1.0 + x) * t);
	double t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_3 = x / (1.0 + x);
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.02) {
		tmp = t_3;
	} else if (t_2 <= 200000000000.0) {
		tmp = 1.0;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / ((1.0 + x) * t)
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	t_3 = x / (1.0 + x)
	tmp = 0
	if t_2 <= -500000.0:
		tmp = t_1
	elif t_2 <= 0.02:
		tmp = t_3
	elif t_2 <= 200000000000.0:
		tmp = 1.0
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(Float64(1.0 + x) * t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_3 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (t_2 <= -500000.0)
		tmp = t_1;
	elseif (t_2 <= 0.02)
		tmp = t_3;
	elseif (t_2 <= 200000000000.0)
		tmp = 1.0;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / ((1.0 + x) * t);
	t_2 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	t_3 = x / (1.0 + x);
	tmp = 0.0;
	if (t_2 <= -500000.0)
		tmp = t_1;
	elseif (t_2 <= 0.02)
		tmp = t_3;
	elseif (t_2 <= 200000000000.0)
		tmp = 1.0;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(N[(1.0 + x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000.0], t$95$1, If[LessEqual[t$95$2, 0.02], t$95$3, If[LessEqual[t$95$2, 200000000000.0], 1.0, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]]

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))) < -5e5 or 2e11 < (/.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 78.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.3%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 63.7

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

   5. Applied rewrites 63.7%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 2e11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.8%

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

Alternative 14: 73.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ t_2 := \frac{x}{1 + x}\\ \mathbf{if}\;t\_1 \leq -500000:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+26}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
        (t_2 (/ x (+ 1.0 x))))
   (if (<= t_1 -500000.0)
     (/ y t)
     (if (<= t_1 0.02)
       t_2
       (if (<= t_1 1e+26) 1.0 (if (<= t_1 INFINITY) (/ y t) t_2))))))

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 = x / (1.0 + x);
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = y / t;
	} else if (t_1 <= 0.02) {
		tmp = t_2;
	} else if (t_1 <= 1e+26) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double t_2 = x / (1.0 + x);
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = y / t;
	} else if (t_1 <= 0.02) {
		tmp = t_2;
	} else if (t_1 <= 1e+26) {
		tmp = 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y / t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)
	t_2 = x / (1.0 + x)
	tmp = 0
	if t_1 <= -500000.0:
		tmp = y / t
	elif t_1 <= 0.02:
		tmp = t_2
	elif t_1 <= 1e+26:
		tmp = 1.0
	elif t_1 <= math.inf:
		tmp = y / t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	t_2 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -500000.0)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.02)
		tmp = t_2;
	elseif (t_1 <= 1e+26)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(y / t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	t_2 = x / (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -500000.0)
		tmp = y / t;
	elseif (t_1 <= 0.02)
		tmp = t_2;
	elseif (t_1 <= 1e+26)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = y / t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -500000.0], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.02], t$95$2, If[LessEqual[t$95$1, 1e+26], 1.0, If[LessEqual[t$95$1, Infinity], N[(y / t), $MachinePrecision], t$95$2]]]]]]

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))) < -5e5 or 1.00000000000000005e26 < (/.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 78.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 63.7

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

   5. Applied rewrites 63.7%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e26

   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 94.4%

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

Alternative 15: 73.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -500000:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 10^{+26}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -500000.0)
     (/ y t)
     (if (<= t_1 0.02)
       (* (fma (- x 1.0) x 1.0) x)
       (if (<= t_1 1e+26) 1.0 (if (<= t_1 INFINITY) (/ y t) 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = y / t;
	} else if (t_1 <= 0.02) {
		tmp = fma((x - 1.0), x, 1.0) * x;
	} else if (t_1 <= 1e+26) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -500000.0)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.02)
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	elseif (t_1 <= 1e+26)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(y / t);
	else
		tmp = 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[LessEqual[t$95$1, -500000.0], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 1e+26], 1.0, If[LessEqual[t$95$1, Infinity], N[(y / t), $MachinePrecision], 1.0]]]]]

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))) < -5e5 or 1.00000000000000005e26 < (/.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 78.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 97.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

   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 59.3%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e26 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 91.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 92.0%

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

Alternative 16: 73.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -500000:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 10^{+26}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 -500000.0)
     (/ y t)
     (if (<= t_1 0.02)
       (* (fma -1.0 x 1.0) x)
       (if (<= t_1 1e+26) 1.0 (if (<= t_1 INFINITY) (/ y t) 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = y / t;
	} else if (t_1 <= 0.02) {
		tmp = fma(-1.0, x, 1.0) * x;
	} else if (t_1 <= 1e+26) {
		tmp = 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -500000.0)
		tmp = Float64(y / t);
	elseif (t_1 <= 0.02)
		tmp = Float64(fma(-1.0, x, 1.0) * x);
	elseif (t_1 <= 1e+26)
		tmp = 1.0;
	elseif (t_1 <= Inf)
		tmp = Float64(y / t);
	else
		tmp = 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[LessEqual[t$95$1, -500000.0], N[(y / t), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(-1.0 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 1e+26], 1.0, If[LessEqual[t$95$1, Infinity], N[(y / t), $MachinePrecision], 1.0]]]]]

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))) < -5e5 or 1.00000000000000005e26 < (/.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 78.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -5e5 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 97.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

   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 59.1%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000005e26 or +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 91.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 92.0%

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

Alternative 17: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t, z, -x\right)} \cdot \frac{z}{1 + x}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (* (/ y (fma t z (- x))) (/ z (+ 1.0 x)))
     (if (<= t_1 2e+271) t_1 (/ (+ (/ y t) x) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / fma(t, z, -x)) * (z / (1.0 + x));
	} else if (t_1 <= 2e+271) {
		tmp = t_1;
	} else {
		tmp = ((y / t) + x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(t, z, Float64(-x))) * Float64(z / Float64(1.0 + x)));
	elseif (t_1 <= 2e+271)
		tmp = t_1;
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision] * N[(z / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+271], t$95$1, N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 38.2%

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

   3. Taylor expanded in y around inf

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

      1. *-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 78.7

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

   5. Applied rewrites 78.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.99999999999999991e271

   1. Initial program 99.4%

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

   if 1.99999999999999991e271 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 32.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 93.0

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

   5. Applied rewrites 93.0%

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

Alternative 18: 62.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) 0.02)
   (* (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)) <= 0.02) {
		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)) <= 0.02)
		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], 0.02], 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))) < 0.0200000000000000004

   1. Initial program 87.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 36.5

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

   5. Applied rewrites 36.5%

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

   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 32.6%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 89.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 73.4%

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

Alternative 19: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2200000000 \lor \neg \left(z \leq 4.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(x, x, x\right)}, -y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2200000000.0) (not (<= z 4.5e+18)))
   (/ (+ (/ y t) x) (+ x 1.0))
   (fma (/ z (fma x x x)) (- y) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2200000000.0) || !(z <= 4.5e+18)) {
		tmp = ((y / t) + x) / (x + 1.0);
	} else {
		tmp = fma((z / fma(x, x, x)), -y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2200000000.0) || !(z <= 4.5e+18))
		tmp = Float64(Float64(Float64(y / t) + x) / Float64(x + 1.0));
	else
		tmp = fma(Float64(z / fma(x, x, x)), Float64(-y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2200000000.0], N[Not[LessEqual[z, 4.5e+18]], $MachinePrecision]], N[(N[(N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(x * x + x), $MachinePrecision]), $MachinePrecision] * (-y) + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e9 or 4.5e18 < z

   1. Initial program 77.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   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 90.1

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

   5. Applied rewrites 90.1%

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

   if -2.2e9 < z < 4.5e18

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. div-add N/A

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

      3. *-inverses N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. associate-/r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 86.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2200000000 \lor \neg \left(z \leq 4.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(x, x, x\right)}, -y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 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 88.5%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 53.0%

   \[\leadsto \color{blue}{1} \]
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 2025006 
(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)))