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

Percentage Accurate: 89.0% → 97.5%

Time: 13.6s

Alternatives: 11

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

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

real(8) function code(x, y, z, t)
    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: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t_1}, x\right)}{x + 1}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{t \cdot \left(z \cdot \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 (- INFINITY))
     (/ (fma z (/ y t_1) x) (+ x 1.0))
     (if (<= t_2 2e+294)
       t_2
       (-
        (+ (/ y (* t (+ x 1.0))) (/ x (+ x 1.0)))
        (/ x (* t (* z (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(z, (y / t_1), x) / (x + 1.0);
	} else if (t_2 <= 2e+294) {
		tmp = t_2;
	} else {
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / (t * (z * (x + 1.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(z, Float64(y / t_1), x) / Float64(x + 1.0));
	elseif (t_2 <= 2e+294)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / Float64(t * Float64(x + 1.0))) + Float64(x / Float64(x + 1.0))) - Float64(x / Float64(t * Float64(z * Float64(x + 1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(z * N[(y / t$95$1), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], t$95$2, N[(N[(N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(t * N[(z * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 37.1%

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

      1. *-commutative 37.1%

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

   3. Simplified 37.1%

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

      1. add-cube-cbrt 37.1%

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

      2. pow3 37.1%

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

      3. fma-neg 37.1%

         \[\leadsto \frac{x + \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(y, z, -x\right)}}\right)}^{3}}{z \cdot t - x}}{x + 1} \]

   5. Applied egg-rr 37.1%

      \[\leadsto \frac{x + \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(y, z, -x\right)}\right)}^{3}}}{z \cdot t - x}}{x + 1} \]

   6. Taylor expanded in x around 0 35.6%

      \[\leadsto \frac{x + \frac{{\color{blue}{\left({\left(y \cdot z\right)}^{0.3333333333333333}\right)}}^{3}}{z \cdot t - x}}{x + 1} \]
   7. Step-by-step derivation

      1. unpow1/3 37.1%

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

      2. *-commutative 37.1%

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

   8. Simplified 37.1%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + \frac{{\left(\sqrt[3]{z \cdot y}\right)}^{3}}{z \cdot t - x}}{x + 1}\right)\right)} \]

      2. expm1-udef 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x + \frac{{\left(\sqrt[3]{z \cdot y}\right)}^{3}}{z \cdot t - x}}{x + 1}\right)} - 1} \]

   10. Applied egg-rr 11.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x + y \cdot \frac{z}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 11.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + y \cdot \frac{z}{\mathsf{fma}\left(z, t, -x\right)}}{x + 1}\right)\right)} \]

       2. expm1-log1p 89.4%

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

       3. +-commutative 89.4%

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

       4. associate-*r/ 37.1%

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

       5. *-commutative 37.1%

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

       6. associate-*r/ 89.4%

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

       7. *-rgt-identity 89.4%

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

       8. associate-*r/ 89.4%

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

       9. fma-def 89.4%

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

       10. associate-*r/ 89.4%

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

       11. *-rgt-identity 89.4%

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

       12. fma-neg 89.4%

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

   12. Simplified 89.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 2.00000000000000013e294

   1. Initial program 99.4%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 21.2%

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

      1. *-commutative 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in t around inf 86.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{z \cdot t - x}, x\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{t \cdot \left(z \cdot \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 2: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{y} \cdot \frac{t_1}{z}}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{t \cdot \left(z \cdot \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -5e+117)
     (/ 1.0 (* (/ (+ x 1.0) y) (/ t_1 z)))
     (if (<= t_2 2e+294)
       t_2
       (-
        (+ (/ y (* t (+ x 1.0))) (/ x (+ x 1.0)))
        (/ x (* t (* z (+ x 1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+117) {
		tmp = 1.0 / (((x + 1.0) / y) * (t_1 / z));
	} else if (t_2 <= 2e+294) {
		tmp = t_2;
	} else {
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / (t * (z * (x + 1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * t) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_2 <= (-5d+117)) then
        tmp = 1.0d0 / (((x + 1.0d0) / y) * (t_1 / z))
    else if (t_2 <= 2d+294) then
        tmp = t_2
    else
        tmp = ((y / (t * (x + 1.0d0))) + (x / (x + 1.0d0))) - (x / (t * (z * (x + 1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+117) {
		tmp = 1.0 / (((x + 1.0) / y) * (t_1 / z));
	} else if (t_2 <= 2e+294) {
		tmp = t_2;
	} else {
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / (t * (z * (x + 1.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -5e+117:
		tmp = 1.0 / (((x + 1.0) / y) * (t_1 / z))
	elif t_2 <= 2e+294:
		tmp = t_2
	else:
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / (t * (z * (x + 1.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e+117)
		tmp = Float64(1.0 / Float64(Float64(Float64(x + 1.0) / y) * Float64(t_1 / z)));
	elseif (t_2 <= 2e+294)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / Float64(t * Float64(x + 1.0))) + Float64(x / Float64(x + 1.0))) - Float64(x / Float64(t * Float64(z * Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -5e+117)
		tmp = 1.0 / (((x + 1.0) / y) * (t_1 / z));
	elseif (t_2 <= 2e+294)
		tmp = t_2;
	else
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / (t * (z * (x + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+117], N[(1.0 / N[(N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision] * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], t$95$2, N[(N[(N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(t * N[(z * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -4.99999999999999983e117

   1. Initial program 69.9%

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

      1. *-commutative 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in y around inf 69.9%

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

      1. clear-num 70.0%

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

      2. inv-pow 70.0%

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

      3. +-commutative 70.0%

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

      4. times-frac 94.6%

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

      5. *-commutative 94.6%

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

      6. fma-neg 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. unpow-1 94.6%

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

      2. fma-neg 94.6%

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

      3. *-commutative 94.6%

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

      4. *-commutative 94.6%

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

   8. Simplified 94.6%

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

   if -4.99999999999999983e117 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 2.00000000000000013e294

   1. Initial program 99.4%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 21.2%

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

      1. *-commutative 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in t around inf 86.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{y} \cdot \frac{z \cdot t - x}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{t \cdot \left(z \cdot \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 3: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{y} \cdot \frac{t_1}{z}}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -5e+117)
     (/ 1.0 (* (/ (+ x 1.0) y) (/ t_1 z)))
     (if (<= t_2 2e+294) t_2 (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+117) {
		tmp = 1.0 / (((x + 1.0) / y) * (t_1 / z));
	} else if (t_2 <= 2e+294) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * t) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_2 <= (-5d+117)) then
        tmp = 1.0d0 / (((x + 1.0d0) / y) * (t_1 / z))
    else if (t_2 <= 2d+294) then
        tmp = t_2
    else
        tmp = (x + (y / t)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+117) {
		tmp = 1.0 / (((x + 1.0) / y) * (t_1 / z));
	} else if (t_2 <= 2e+294) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -5e+117:
		tmp = 1.0 / (((x + 1.0) / y) * (t_1 / z))
	elif t_2 <= 2e+294:
		tmp = t_2
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e+117)
		tmp = Float64(1.0 / Float64(Float64(Float64(x + 1.0) / y) * Float64(t_1 / z)));
	elseif (t_2 <= 2e+294)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -5e+117)
		tmp = 1.0 / (((x + 1.0) / y) * (t_1 / z));
	elseif (t_2 <= 2e+294)
		tmp = t_2;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+117], N[(1.0 / N[(N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision] * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -4.99999999999999983e117

   1. Initial program 69.9%

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

      1. *-commutative 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in y around inf 69.9%

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

      1. clear-num 70.0%

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

      2. inv-pow 70.0%

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

      3. +-commutative 70.0%

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

      4. times-frac 94.6%

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

      5. *-commutative 94.6%

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

      6. fma-neg 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. unpow-1 94.6%

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

      2. fma-neg 94.6%

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

      3. *-commutative 94.6%

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

      4. *-commutative 94.6%

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

   8. Simplified 94.6%

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

   if -4.99999999999999983e117 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 2.00000000000000013e294

   1. Initial program 99.4%

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

   if 2.00000000000000013e294 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 21.2%

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

      1. *-commutative 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{y} \cdot \frac{z \cdot t - x}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 4: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{z}{\frac{x + 1}{\frac{y - t}{x}}}\\ t_2 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-121}:\\ \;\;\;\;z \cdot \frac{\frac{y}{x + 1}}{z \cdot t - x}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ z (/ (+ x 1.0) (/ (- y t) x)))))
        (t_2 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t -2.5e-96)
     t_2
     (if (<= t 9.4e-153)
       t_1
       (if (<= t 1.35e-121)
         (* z (/ (/ y (+ x 1.0)) (- (* z t) x)))
         (if (<= t 4.3e-88) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (z / ((x + 1.0) / ((y - t) / x)));
	double t_2 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t <= -2.5e-96) {
		tmp = t_2;
	} else if (t <= 9.4e-153) {
		tmp = t_1;
	} else if (t <= 1.35e-121) {
		tmp = z * ((y / (x + 1.0)) / ((z * t) - x));
	} else if (t <= 4.3e-88) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 - (z / ((x + 1.0d0) / ((y - t) / x)))
    t_2 = (x + (y / t)) / (x + 1.0d0)
    if (t <= (-2.5d-96)) then
        tmp = t_2
    else if (t <= 9.4d-153) then
        tmp = t_1
    else if (t <= 1.35d-121) then
        tmp = z * ((y / (x + 1.0d0)) / ((z * t) - x))
    else if (t <= 4.3d-88) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (z / ((x + 1.0) / ((y - t) / x)));
	double t_2 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t <= -2.5e-96) {
		tmp = t_2;
	} else if (t <= 9.4e-153) {
		tmp = t_1;
	} else if (t <= 1.35e-121) {
		tmp = z * ((y / (x + 1.0)) / ((z * t) - x));
	} else if (t <= 4.3e-88) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - (z / ((x + 1.0) / ((y - t) / x)))
	t_2 = (x + (y / t)) / (x + 1.0)
	tmp = 0
	if t <= -2.5e-96:
		tmp = t_2
	elif t <= 9.4e-153:
		tmp = t_1
	elif t <= 1.35e-121:
		tmp = z * ((y / (x + 1.0)) / ((z * t) - x))
	elif t <= 4.3e-88:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(z / Float64(Float64(x + 1.0) / Float64(Float64(y - t) / x))))
	t_2 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	tmp = 0.0
	if (t <= -2.5e-96)
		tmp = t_2;
	elseif (t <= 9.4e-153)
		tmp = t_1;
	elseif (t <= 1.35e-121)
		tmp = Float64(z * Float64(Float64(y / Float64(x + 1.0)) / Float64(Float64(z * t) - x)));
	elseif (t <= 4.3e-88)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - (z / ((x + 1.0) / ((y - t) / x)));
	t_2 = (x + (y / t)) / (x + 1.0);
	tmp = 0.0;
	if (t <= -2.5e-96)
		tmp = t_2;
	elseif (t <= 9.4e-153)
		tmp = t_1;
	elseif (t <= 1.35e-121)
		tmp = z * ((y / (x + 1.0)) / ((z * t) - x));
	elseif (t <= 4.3e-88)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(z / N[(N[(x + 1.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.5e-96], t$95$2, If[LessEqual[t, 9.4e-153], t$95$1, If[LessEqual[t, 1.35e-121], N[(z * N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e-88], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.49999999999999997e-96 or 4.2999999999999997e-88 < t

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in z around inf 89.5%

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

   if -2.49999999999999997e-96 < t < 9.3999999999999998e-153 or 1.3500000000000001e-121 < t < 4.2999999999999997e-88

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in z around 0 77.3%

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

      1. distribute-lft-out-- 77.3%

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

   6. Simplified 77.3%

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

   7. Taylor expanded in z around 0 77.3%

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

      1. mul-1-neg 77.3%

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

      2. unsub-neg 77.3%

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

      3. div-sub 77.3%

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

      4. associate-/l* 79.2%

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

      5. +-commutative 79.2%

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

   9. Simplified 79.2%

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

   if 9.3999999999999998e-153 < t < 1.3500000000000001e-121

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

   3. Simplified 62.3%

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

      1. add-cube-cbrt 61.7%

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

      2. pow3 61.7%

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

      3. fma-neg 61.7%

         \[\leadsto \frac{x + \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(y, z, -x\right)}}\right)}^{3}}{z \cdot t - x}}{x + 1} \]

   5. Applied egg-rr 61.7%

      \[\leadsto \frac{x + \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(y, z, -x\right)}\right)}^{3}}}{z \cdot t - x}}{x + 1} \]

   6. Taylor expanded in x around 0 38.8%

      \[\leadsto \frac{x + \frac{{\color{blue}{\left({\left(y \cdot z\right)}^{0.3333333333333333}\right)}}^{3}}{z \cdot t - x}}{x + 1} \]
   7. Step-by-step derivation

      1. unpow1/3 61.7%

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

      2. *-commutative 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in y around inf 61.2%

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

       1. associate-/r* 62.3%

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

       2. +-commutative 62.3%

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

       3. associate-*l/ 62.3%

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

       4. *-commutative 62.3%

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

       5. *-commutative 62.3%

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

       6. fma-neg 62.3%

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

       7. associate-*r/ 99.4%

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

       8. fma-neg 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{-153}:\\ \;\;\;\;1 - \frac{z}{\frac{x + 1}{\frac{y - t}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-121}:\\ \;\;\;\;z \cdot \frac{\frac{y}{x + 1}}{z \cdot t - x}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-88}:\\ \;\;\;\;1 - \frac{z}{\frac{x + 1}{\frac{y - t}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 5: 81.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(x + 1\right) - \frac{y}{\frac{x}{z}}}{x + 1}\\ t_2 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-121}:\\ \;\;\;\;z \cdot \frac{\frac{y}{x + 1}}{z \cdot t - x}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- (+ x 1.0) (/ y (/ x z))) (+ x 1.0)))
        (t_2 (/ (+ x (/ y t)) (+ x 1.0))))
   (if (<= t -2.55e-98)
     t_2
     (if (<= t 1e-152)
       t_1
       (if (<= t 1.45e-121)
         (* z (/ (/ y (+ x 1.0)) (- (* z t) x)))
         (if (<= t 1.22e-88) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x + 1.0) - (y / (x / z))) / (x + 1.0);
	double t_2 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t <= -2.55e-98) {
		tmp = t_2;
	} else if (t <= 1e-152) {
		tmp = t_1;
	} else if (t <= 1.45e-121) {
		tmp = z * ((y / (x + 1.0)) / ((z * t) - x));
	} else if (t <= 1.22e-88) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x + 1.0d0) - (y / (x / z))) / (x + 1.0d0)
    t_2 = (x + (y / t)) / (x + 1.0d0)
    if (t <= (-2.55d-98)) then
        tmp = t_2
    else if (t <= 1d-152) then
        tmp = t_1
    else if (t <= 1.45d-121) then
        tmp = z * ((y / (x + 1.0d0)) / ((z * t) - x))
    else if (t <= 1.22d-88) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x + 1.0) - (y / (x / z))) / (x + 1.0);
	double t_2 = (x + (y / t)) / (x + 1.0);
	double tmp;
	if (t <= -2.55e-98) {
		tmp = t_2;
	} else if (t <= 1e-152) {
		tmp = t_1;
	} else if (t <= 1.45e-121) {
		tmp = z * ((y / (x + 1.0)) / ((z * t) - x));
	} else if (t <= 1.22e-88) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x + 1.0) - (y / (x / z))) / (x + 1.0)
	t_2 = (x + (y / t)) / (x + 1.0)
	tmp = 0
	if t <= -2.55e-98:
		tmp = t_2
	elif t <= 1e-152:
		tmp = t_1
	elif t <= 1.45e-121:
		tmp = z * ((y / (x + 1.0)) / ((z * t) - x))
	elif t <= 1.22e-88:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x + 1.0) - Float64(y / Float64(x / z))) / Float64(x + 1.0))
	t_2 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	tmp = 0.0
	if (t <= -2.55e-98)
		tmp = t_2;
	elseif (t <= 1e-152)
		tmp = t_1;
	elseif (t <= 1.45e-121)
		tmp = Float64(z * Float64(Float64(y / Float64(x + 1.0)) / Float64(Float64(z * t) - x)));
	elseif (t <= 1.22e-88)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x + 1.0) - (y / (x / z))) / (x + 1.0);
	t_2 = (x + (y / t)) / (x + 1.0);
	tmp = 0.0;
	if (t <= -2.55e-98)
		tmp = t_2;
	elseif (t <= 1e-152)
		tmp = t_1;
	elseif (t <= 1.45e-121)
		tmp = z * ((y / (x + 1.0)) / ((z * t) - x));
	elseif (t <= 1.22e-88)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x + 1.0), $MachinePrecision] - N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.55e-98], t$95$2, If[LessEqual[t, 1e-152], t$95$1, If[LessEqual[t, 1.45e-121], N[(z * N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.22e-88], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.55000000000000011e-98 or 1.2200000000000001e-88 < t

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in z around inf 89.5%

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

   if -2.55000000000000011e-98 < t < 1.00000000000000007e-152 or 1.45e-121 < t < 1.2200000000000001e-88

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in t around 0 78.5%

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

      1. associate-+r+ 78.5%

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

      2. mul-1-neg 78.5%

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

      3. unsub-neg 78.5%

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

      4. +-commutative 78.5%

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

      5. associate-/l* 80.9%

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

      6. +-commutative 80.9%

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

   6. Simplified 80.9%

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

   if 1.00000000000000007e-152 < t < 1.45e-121

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

   3. Simplified 62.3%

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

      1. add-cube-cbrt 61.7%

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

      2. pow3 61.7%

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

      3. fma-neg 61.7%

         \[\leadsto \frac{x + \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(y, z, -x\right)}}\right)}^{3}}{z \cdot t - x}}{x + 1} \]

   5. Applied egg-rr 61.7%

      \[\leadsto \frac{x + \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(y, z, -x\right)}\right)}^{3}}}{z \cdot t - x}}{x + 1} \]

   6. Taylor expanded in x around 0 38.8%

      \[\leadsto \frac{x + \frac{{\color{blue}{\left({\left(y \cdot z\right)}^{0.3333333333333333}\right)}}^{3}}{z \cdot t - x}}{x + 1} \]
   7. Step-by-step derivation

      1. unpow1/3 61.7%

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

      2. *-commutative 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in y around inf 61.2%

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

       1. associate-/r* 62.3%

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

       2. +-commutative 62.3%

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

       3. associate-*l/ 62.3%

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

       4. *-commutative 62.3%

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

       5. *-commutative 62.3%

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

       6. fma-neg 62.3%

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

       7. associate-*r/ 99.4%

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

       8. fma-neg 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{-98}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;t \leq 10^{-152}:\\ \;\;\;\;\frac{\left(x + 1\right) - \frac{y}{\frac{x}{z}}}{x + 1}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-121}:\\ \;\;\;\;z \cdot \frac{\frac{y}{x + 1}}{z \cdot t - x}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(x + 1\right) - \frac{y}{\frac{x}{z}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 6: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-35} \lor \neg \left(t \leq 4.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{z \cdot t - x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.3e-35) (not (<= t 4.3e-47)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (/ (+ x (/ (* y z) (- (* z t) x))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.3e-35) || !(t <= 4.3e-47)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x + ((y * z) / ((z * t) - x))) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-4.3d-35)) .or. (.not. (t <= 4.3d-47))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = (x + ((y * z) / ((z * t) - x))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.3e-35) || !(t <= 4.3e-47)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x + ((y * z) / ((z * t) - x))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.3e-35) or not (t <= 4.3e-47):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = (x + ((y * z) / ((z * t) - x))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.3e-35) || !(t <= 4.3e-47))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(y * z) / Float64(Float64(z * t) - x))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.3e-35) || ~((t <= 4.3e-47)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = (x + ((y * z) / ((z * t) - x))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.3e-35], N[Not[LessEqual[t, 4.3e-47]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(y * z), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3000000000000002e-35 or 4.2999999999999998e-47 < t

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in z around inf 90.4%

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

   if -4.3000000000000002e-35 < t < 4.2999999999999998e-47

   1. Initial program 93.3%

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

      1. *-commutative 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in y around inf 86.7%

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

      1. *-commutative 86.7%

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

   6. Simplified 86.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-35} \lor \neg \left(t \leq 4.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{z \cdot t - x}}{x + 1}\\ \end{array} \]

Alternative 7: 77.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-158} \lor \neg \left(t \leq 3.7 \cdot 10^{-163}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{y}{\frac{x}{z}}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.4e-158) (not (<= t 3.7e-163)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (/ (- x (/ y (/ x z))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.4e-158) || !(t <= 3.7e-163)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x - (y / (x / z))) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-4.4d-158)) .or. (.not. (t <= 3.7d-163))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = (x - (y / (x / z))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.4e-158) || !(t <= 3.7e-163)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x - (y / (x / z))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.4e-158) or not (t <= 3.7e-163):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = (x - (y / (x / z))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.4e-158) || !(t <= 3.7e-163))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x - Float64(y / Float64(x / z))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.4e-158) || ~((t <= 3.7e-163)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = (x - (y / (x / z))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.4e-158], N[Not[LessEqual[t, 3.7e-163]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.4000000000000002e-158 or 3.6999999999999999e-163 < t

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in z around inf 84.7%

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

   if -4.4000000000000002e-158 < t < 3.6999999999999999e-163

   1. Initial program 93.3%

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

      1. *-commutative 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in y around inf 86.8%

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

      1. *-commutative 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in t around 0 73.5%

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

      1. mul-1-neg 73.5%

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

      2. unsub-neg 73.5%

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

      3. associate-/l* 76.7%

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

      4. +-commutative 76.7%

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

   9. Simplified 76.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-158} \lor \neg \left(t \leq 3.7 \cdot 10^{-163}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{y}{\frac{x}{z}}}{x + 1}\\ \end{array} \]

Alternative 8: 77.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.2e+25)
   1.0
   (if (<= x 3.4e+30) (/ (+ x (/ y t)) (+ x 1.0)) (+ 1.0 (/ -1.0 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.2e+25) {
		tmp = 1.0;
	} else if (x <= 3.4e+30) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-5.2d+25)) then
        tmp = 1.0d0
    else if (x <= 3.4d+30) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.2e+25) {
		tmp = 1.0;
	} else if (x <= 3.4e+30) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.2e+25:
		tmp = 1.0
	elif x <= 3.4e+30:
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.2e+25)
		tmp = 1.0;
	elseif (x <= 3.4e+30)
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -5.2e+25)
		tmp = 1.0;
	elseif (x <= 3.4e+30)
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.2e+25], 1.0, If[LessEqual[x, 3.4e+30], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999997e25

   1. Initial program 91.9%

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

      1. *-commutative 91.9%

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

   3. Simplified 91.9%

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

      1. add-cube-cbrt 91.8%

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

      2. pow3 91.8%

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

      3. fma-neg 91.8%

         \[\leadsto \frac{x + \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(y, z, -x\right)}}\right)}^{3}}{z \cdot t - x}}{x + 1} \]

   5. Applied egg-rr 91.8%

      \[\leadsto \frac{x + \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(y, z, -x\right)}\right)}^{3}}}{z \cdot t - x}}{x + 1} \]

   6. Taylor expanded in x around 0 43.2%

      \[\leadsto \frac{x + \frac{{\color{blue}{\left({\left(y \cdot z\right)}^{0.3333333333333333}\right)}}^{3}}{z \cdot t - x}}{x + 1} \]
   7. Step-by-step derivation

      1. unpow1/3 91.8%

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

      2. *-commutative 91.8%

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

   8. Simplified 91.8%

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

   9. Taylor expanded in x around inf 93.5%

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

   if -5.1999999999999997e25 < x < 3.4000000000000002e30

   1. Initial program 92.1%

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

      1. *-commutative 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in z around inf 70.4%

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

   if 3.4000000000000002e30 < x

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in t around inf 88.2%

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

      1. +-commutative 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in x around inf 88.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 9: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-103} \lor \neg \left(x \leq 1.2 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e-103) (not (<= x 1.2e-119))) (/ x (+ x 1.0)) (/ y t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e-103) || !(x <= 1.2e-119)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2d-103)) .or. (.not. (x <= 1.2d-119))) then
        tmp = x / (x + 1.0d0)
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e-103) || !(x <= 1.2e-119)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2e-103) or not (x <= 1.2e-119):
		tmp = x / (x + 1.0)
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e-103) || !(x <= 1.2e-119))
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2e-103) || ~((x <= 1.2e-119)))
		tmp = x / (x + 1.0);
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e-103], N[Not[LessEqual[x, 1.2e-119]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999992e-103 or 1.20000000000000004e-119 < x

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in t around inf 76.5%

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

      1. +-commutative 76.5%

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

   6. Simplified 76.5%

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

   if -1.99999999999999992e-103 < x < 1.20000000000000004e-119

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in y around inf 68.2%

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

      1. times-frac 72.0%

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

      2. +-commutative 72.0%

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

      3. fma-neg 72.0%

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

   6. Simplified 72.0%

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

   7. Taylor expanded in z around inf 62.0%

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

      1. un-div-inv 62.3%

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

   9. Applied egg-rr 62.3%

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

   10. Taylor expanded in x around 0 62.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-103} \lor \neg \left(x \leq 1.2 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]

Alternative 10: 67.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.9e-89) 1.0 (if (<= x 4e-20) (/ y t) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.9e-89) {
		tmp = 1.0;
	} else if (x <= 4e-20) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-5.9d-89)) then
        tmp = 1.0d0
    else if (x <= 4d-20) then
        tmp = y / t
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.9e-89) {
		tmp = 1.0;
	} else if (x <= 4e-20) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.9e-89:
		tmp = 1.0
	elif x <= 4e-20:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.9e-89)
		tmp = 1.0;
	elseif (x <= 4e-20)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -5.9e-89)
		tmp = 1.0;
	elseif (x <= 4e-20)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.9e-89], 1.0, If[LessEqual[x, 4e-20], N[(y / t), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -5.90000000000000021e-89 or 3.99999999999999978e-20 < x

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

   3. Simplified 90.7%

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

      1. add-cube-cbrt 90.4%

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

      2. pow3 90.4%

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

      3. fma-neg 90.4%

         \[\leadsto \frac{x + \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(y, z, -x\right)}}\right)}^{3}}{z \cdot t - x}}{x + 1} \]

   5. Applied egg-rr 90.4%

      \[\leadsto \frac{x + \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(y, z, -x\right)}\right)}^{3}}}{z \cdot t - x}}{x + 1} \]

   6. Taylor expanded in x around 0 47.9%

      \[\leadsto \frac{x + \frac{{\color{blue}{\left({\left(y \cdot z\right)}^{0.3333333333333333}\right)}}^{3}}{z \cdot t - x}}{x + 1} \]
   7. Step-by-step derivation

      1. unpow1/3 85.6%

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

      2. *-commutative 85.6%

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

   8. Simplified 85.6%

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

   9. Taylor expanded in x around inf 79.3%

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

   if -5.90000000000000021e-89 < x < 3.99999999999999978e-20

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in y around inf 61.4%

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

      1. times-frac 65.3%

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

      2. +-commutative 65.3%

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

      3. fma-neg 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in z around inf 55.5%

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

      1. un-div-inv 55.7%

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

   9. Applied egg-rr 55.7%

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

   10. Taylor expanded in x around 0 55.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 53.3% accurate, 17.0× 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

real(8) function code(x, y, z, t)
    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 90.8%

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

   1. *-commutative 90.8%

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

3. Simplified 90.8%

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

   1. add-cube-cbrt 90.1%

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

   2. pow3 90.1%

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

   3. fma-neg 90.1%

      \[\leadsto \frac{x + \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(y, z, -x\right)}}\right)}^{3}}{z \cdot t - x}}{x + 1} \]

5. Applied egg-rr 90.1%

   \[\leadsto \frac{x + \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(y, z, -x\right)}\right)}^{3}}}{z \cdot t - x}}{x + 1} \]

6. Taylor expanded in x around 0 42.0%

   \[\leadsto \frac{x + \frac{{\color{blue}{\left({\left(y \cdot z\right)}^{0.3333333333333333}\right)}}^{3}}{z \cdot t - x}}{x + 1} \]
7. Step-by-step derivation

   1. unpow1/3 82.1%

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

   2. *-commutative 82.1%

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

8. Simplified 82.1%

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

9. Taylor expanded in x around inf 53.6%

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

10. Final simplification 53.6%

    \[\leadsto 1 \]

Developer target: 99.5% accurate, 0.8× 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

real(8) function code(x, y, z, t)
    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 2023310 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

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