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

Percentage Accurate: 88.9% → 96.1%

Time: 13.3s

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: 88.9% 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: 96.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(z, y, -x\right)}{t\_1}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\right) - \frac{x}{t \cdot \left(z \cdot \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

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 -1e+112)
     (* (/ y (+ x 1.0)) (/ z t_1))
     (if (<= t_2 5e+240)
       (/ (+ x (/ (fma z y (- x)) t_1)) (+ x 1.0))
       (-
        (+ (/ x (+ x 1.0)) (/ y (* t (+ 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 <= -1e+112) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 5e+240) {
		tmp = (x + (fma(z, y, -x) / t_1)) / (x + 1.0);
	} else {
		tmp = ((x / (x + 1.0)) + (y / (t * (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 <= -1e+112)
		tmp = Float64(Float64(y / Float64(x + 1.0)) * Float64(z / t_1));
	elseif (t_2 <= 5e+240)
		tmp = Float64(Float64(x + Float64(fma(z, y, Float64(-x)) / t_1)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * 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, -1e+112], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+240], N[(N[(x + N[(N[(z * y + (-x)), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $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 #s(literal 1 binary64))) < -9.9999999999999993e111

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in y around inf 73.3%

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

      1. times-frac 92.3%

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

      2. +-commutative 92.3%

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

   7. Simplified 92.3%

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

   if -9.9999999999999993e111 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000003e240

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. fmm-def 99.4%

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

   6. Applied egg-rr 99.4%

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

   if 5.0000000000000003e240 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in t around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. +-commutative 84.9%

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

      3. +-commutative 84.9%

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

   7. Simplified 84.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{z \cdot t - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(z, y, -x\right)}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\right) - \frac{x}{t \cdot \left(z \cdot \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+240}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\right) - \frac{x}{t \cdot \left(z \cdot \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

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 -1e+112)
     (* (/ y (+ x 1.0)) (/ z t_1))
     (if (<= t_2 5e+240)
       t_2
       (-
        (+ (/ x (+ x 1.0)) (/ y (* t (+ 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 <= -1e+112) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 5e+240) {
		tmp = t_2;
	} else {
		tmp = ((x / (x + 1.0)) + (y / (t * (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 <= (-1d+112)) then
        tmp = (y / (x + 1.0d0)) * (z / t_1)
    else if (t_2 <= 5d+240) then
        tmp = t_2
    else
        tmp = ((x / (x + 1.0d0)) + (y / (t * (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 <= -1e+112) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 5e+240) {
		tmp = t_2;
	} else {
		tmp = ((x / (x + 1.0)) + (y / (t * (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 <= -1e+112:
		tmp = (y / (x + 1.0)) * (z / t_1)
	elif t_2 <= 5e+240:
		tmp = t_2
	else:
		tmp = ((x / (x + 1.0)) + (y / (t * (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 <= -1e+112)
		tmp = Float64(Float64(y / Float64(x + 1.0)) * Float64(z / t_1));
	elseif (t_2 <= 5e+240)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * 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 <= -1e+112)
		tmp = (y / (x + 1.0)) * (z / t_1);
	elseif (t_2 <= 5e+240)
		tmp = t_2;
	else
		tmp = ((x / (x + 1.0)) + (y / (t * (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, -1e+112], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+240], t$95$2, N[(N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $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 #s(literal 1 binary64))) < -9.9999999999999993e111

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in y around inf 73.3%

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

      1. times-frac 92.3%

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

      2. +-commutative 92.3%

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

   7. Simplified 92.3%

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

   if -9.9999999999999993e111 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000003e240

   1. Initial program 99.4%

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

   if 5.0000000000000003e240 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in t around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. +-commutative 84.9%

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

      3. +-commutative 84.9%

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

   7. Simplified 84.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{z \cdot t - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\right) - \frac{x}{t \cdot \left(z \cdot \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+240}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

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 -1e+112)
     (* (/ y (+ x 1.0)) (/ z t_1))
     (if (<= t_2 5e+240) 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 <= -1e+112) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 5e+240) {
		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 <= (-1d+112)) then
        tmp = (y / (x + 1.0d0)) * (z / t_1)
    else if (t_2 <= 5d+240) 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 <= -1e+112) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 5e+240) {
		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 <= -1e+112:
		tmp = (y / (x + 1.0)) * (z / t_1)
	elif t_2 <= 5e+240:
		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 <= -1e+112)
		tmp = Float64(Float64(y / Float64(x + 1.0)) * Float64(z / t_1));
	elseif (t_2 <= 5e+240)
		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 <= -1e+112)
		tmp = (y / (x + 1.0)) * (z / t_1);
	elseif (t_2 <= 5e+240)
		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, -1e+112], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+240], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999993e111

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in y around inf 73.3%

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

      1. times-frac 92.3%

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

      2. +-commutative 92.3%

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

   7. Simplified 92.3%

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

   if -9.9999999999999993e111 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000003e240

   1. Initial program 99.4%

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

   if 5.0000000000000003e240 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in z around inf 84.8%

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

      1. +-commutative 84.8%

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

      2. +-commutative 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{z \cdot t - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-5} \lor \neg \left(t \leq 0.076\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y \cdot \frac{z}{x}\right) + 1}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.6e-5) (not (<= t 0.076)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (/ (+ (- x (* y (/ z x))) 1.0) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.6e-5) || !(t <= 0.076)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = ((x - (y * (z / x))) + 1.0) / (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 <= (-1.6d-5)) .or. (.not. (t <= 0.076d0))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = ((x - (y * (z / x))) + 1.0d0) / (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 <= -1.6e-5) || !(t <= 0.076)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = ((x - (y * (z / x))) + 1.0) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.6e-5) or not (t <= 0.076):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = ((x - (y * (z / x))) + 1.0) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.6e-5) || !(t <= 0.076))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(x - Float64(y * Float64(z / x))) + 1.0) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.6e-5) || ~((t <= 0.076)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = ((x - (y * (z / x))) + 1.0) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.6e-5], N[Not[LessEqual[t, 0.076]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.59999999999999993e-5 or 0.0759999999999999981 < t

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in z around inf 91.4%

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

      1. +-commutative 91.4%

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

      2. +-commutative 91.4%

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

   7. Simplified 91.4%

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

   if -1.59999999999999993e-5 < t < 0.0759999999999999981

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in t around 0 77.1%

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

      1. mul-1-neg 77.1%

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

      2. unsub-neg 77.1%

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

      3. associate-/l* 81.9%

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

      4. +-commutative 81.9%

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

   7. Simplified 81.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-5} \lor \neg \left(t \leq 0.076\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y \cdot \frac{z}{x}\right) + 1}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{z}{x \cdot \left(-1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.5e-9)
   1.0
   (if (<= x 5.6e-33)
     (/ (+ x (/ (- y (/ x z)) t)) (+ x 1.0))
     (+ 1.0 (* y (/ z (* x (- -1.0 x))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.5e-9) {
		tmp = 1.0;
	} else if (x <= 5.6e-33) {
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	} else {
		tmp = 1.0 + (y * (z / (x * (-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 <= (-3.5d-9)) then
        tmp = 1.0d0
    else if (x <= 5.6d-33) then
        tmp = (x + ((y - (x / z)) / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 + (y * (z / (x * ((-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 <= -3.5e-9) {
		tmp = 1.0;
	} else if (x <= 5.6e-33) {
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	} else {
		tmp = 1.0 + (y * (z / (x * (-1.0 - x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.5e-9:
		tmp = 1.0
	elif x <= 5.6e-33:
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0)
	else:
		tmp = 1.0 + (y * (z / (x * (-1.0 - x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.5e-9)
		tmp = 1.0;
	elseif (x <= 5.6e-33)
		tmp = Float64(Float64(x + Float64(Float64(y - Float64(x / z)) / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 + Float64(y * Float64(z / Float64(x * Float64(-1.0 - x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.5e-9)
		tmp = 1.0;
	elseif (x <= 5.6e-33)
		tmp = (x + ((y - (x / z)) / t)) / (x + 1.0);
	else
		tmp = 1.0 + (y * (z / (x * (-1.0 - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.5e-9], 1.0, If[LessEqual[x, 5.6e-33], N[(N[(x + N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(z / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999999e-9

   1. Initial program 89.1%

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

      1. *-commutative 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around inf 94.7%

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

   if -3.4999999999999999e-9 < x < 5.6e-33

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in t around -inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. unsub-neg 75.5%

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

      3. cancel-sign-sub-inv 75.5%

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

      4. metadata-eval 75.5%

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

      5. *-lft-identity 75.5%

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

      6. +-commutative 75.5%

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

      7. mul-1-neg 75.5%

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

      8. unsub-neg 75.5%

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

   7. Simplified 75.5%

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

   if 5.6e-33 < x

   1. Initial program 88.2%

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

      1. *-commutative 88.2%

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

   3. Simplified 88.2%

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

      1. clear-num 88.1%

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

      2. inv-pow 88.1%

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

   6. Applied egg-rr 88.1%

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

      1. unpow-1 88.1%

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

      2. *-commutative 88.1%

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

   8. Simplified 88.1%

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

   9. Taylor expanded in t around 0 84.3%

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

       1. +-commutative 84.3%

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

       2. associate-+r+ 84.3%

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

       3. +-commutative 84.3%

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

       4. mul-1-neg 84.3%

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

       5. associate-/l* 90.2%

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

   11. Simplified 90.2%

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

   12. Taylor expanded in y around 0 84.1%

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

       1. mul-1-neg 84.1%

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

       2. times-frac 91.4%

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

       3. distribute-lft-neg-in 91.4%

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

       4. cancel-sign-sub-inv 91.4%

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

       5. times-frac 84.1%

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

       6. associate-/l* 94.2%

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

       7. +-commutative 94.2%

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

   14. Simplified 94.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{x + \frac{y - \frac{x}{z}}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{z}{x \cdot \left(-1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0015 \lor \neg \left(t \leq 8.5 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{z}{x \cdot \left(-1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -0.0015) (not (<= t 8.5e-32)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (+ 1.0 (* y (/ z (* x (- -1.0 x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.0015) || !(t <= 8.5e-32)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 + (y * (z / (x * (-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 ((t <= (-0.0015d0)) .or. (.not. (t <= 8.5d-32))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 + (y * (z / (x * ((-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 ((t <= -0.0015) || !(t <= 8.5e-32)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 + (y * (z / (x * (-1.0 - x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -0.0015) or not (t <= 8.5e-32):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 + (y * (z / (x * (-1.0 - x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -0.0015) || !(t <= 8.5e-32))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 + Float64(y * Float64(z / Float64(x * Float64(-1.0 - x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -0.0015) || ~((t <= 8.5e-32)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 + (y * (z / (x * (-1.0 - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -0.0015], N[Not[LessEqual[t, 8.5e-32]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(z / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0015 or 8.5000000000000003e-32 < t

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in z around inf 90.3%

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

      1. +-commutative 90.3%

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

      2. +-commutative 90.3%

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

   7. Simplified 90.3%

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

   if -0.0015 < t < 8.5000000000000003e-32

   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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 93.3%

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

      2. inv-pow 93.3%

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

   6. Applied egg-rr 93.3%

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

      1. unpow-1 93.3%

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

      2. *-commutative 93.3%

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

   8. Simplified 93.3%

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

   9. Taylor expanded in t around 0 77.3%

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

       1. +-commutative 77.3%

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

       2. associate-+r+ 77.3%

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

       3. +-commutative 77.3%

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

       4. mul-1-neg 77.3%

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

       5. associate-/l* 82.5%

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

   11. Simplified 82.5%

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

   12. Taylor expanded in y around 0 77.2%

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

       1. mul-1-neg 77.2%

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

       2. times-frac 79.6%

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

       3. distribute-lft-neg-in 79.6%

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

       4. cancel-sign-sub-inv 79.6%

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

       5. times-frac 77.2%

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

       6. associate-/l* 82.4%

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

       7. +-commutative 82.4%

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

   14. Simplified 82.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0015 \lor \neg \left(t \leq 8.5 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{z}{x \cdot \left(-1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.12 \cdot 10^{-32}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.2e-9) 1.0 (if (<= x 2.12e-32) (/ (+ x (/ y t)) (+ x 1.0)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.2e-9) {
		tmp = 1.0;
	} else if (x <= 2.12e-32) {
		tmp = (x + (y / t)) / (x + 1.0);
	} 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 <= (-1.2d-9)) then
        tmp = 1.0d0
    else if (x <= 2.12d-32) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.2e-9) {
		tmp = 1.0;
	} else if (x <= 2.12e-32) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.2e-9:
		tmp = 1.0
	elif x <= 2.12e-32:
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.2e-9)
		tmp = 1.0;
	elseif (x <= 2.12e-32)
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.2e-9)
		tmp = 1.0;
	elseif (x <= 2.12e-32)
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.2e-9], 1.0, If[LessEqual[x, 2.12e-32], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-9 or 2.11999999999999987e-32 < x

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in x around inf 91.6%

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

   if -1.2e-9 < x < 2.11999999999999987e-32

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in z around inf 72.4%

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

      1. +-commutative 72.4%

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

      2. +-commutative 72.4%

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

   7. Simplified 72.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.12 \cdot 10^{-32}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot t - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7.2e-18) 1.0 (if (<= x 6.6e-19) (* y (/ z (- (* z t) x))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.2e-18) {
		tmp = 1.0;
	} else if (x <= 6.6e-19) {
		tmp = y * (z / ((z * t) - x));
	} 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 <= (-7.2d-18)) then
        tmp = 1.0d0
    else if (x <= 6.6d-19) then
        tmp = y * (z / ((z * t) - x))
    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 <= -7.2e-18) {
		tmp = 1.0;
	} else if (x <= 6.6e-19) {
		tmp = y * (z / ((z * t) - x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -7.2e-18:
		tmp = 1.0
	elif x <= 6.6e-19:
		tmp = y * (z / ((z * t) - x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7.2e-18)
		tmp = 1.0;
	elseif (x <= 6.6e-19)
		tmp = Float64(y * Float64(z / Float64(Float64(z * t) - x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -7.2e-18)
		tmp = 1.0;
	elseif (x <= 6.6e-19)
		tmp = y * (z / ((z * t) - x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7.2e-18], 1.0, If[LessEqual[x, 6.6e-19], N[(y * N[(z / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000021e-18 or 6.5999999999999995e-19 < x

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in x around inf 92.7%

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

   if -7.20000000000000021e-18 < x < 6.5999999999999995e-19

   1. Initial program 87.6%

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

      1. *-commutative 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in y around inf 54.6%

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

      1. times-frac 58.4%

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

      2. +-commutative 58.4%

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

   7. Simplified 58.4%

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

   8. Taylor expanded in x around 0 58.4%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot t - x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.6e-22) 1.0 (if (<= x 5e-37) (/ y (* t (+ x 1.0))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.6e-22) {
		tmp = 1.0;
	} else if (x <= 5e-37) {
		tmp = y / (t * (x + 1.0));
	} 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 <= (-3.6d-22)) then
        tmp = 1.0d0
    else if (x <= 5d-37) then
        tmp = y / (t * (x + 1.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.6e-22) {
		tmp = 1.0;
	} else if (x <= 5e-37) {
		tmp = y / (t * (x + 1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.6e-22:
		tmp = 1.0
	elif x <= 5e-37:
		tmp = y / (t * (x + 1.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.6e-22)
		tmp = 1.0;
	elseif (x <= 5e-37)
		tmp = Float64(y / Float64(t * Float64(x + 1.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.6e-22)
		tmp = 1.0;
	elseif (x <= 5e-37)
		tmp = y / (t * (x + 1.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.6e-22], 1.0, If[LessEqual[x, 5e-37], N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999998e-22 or 4.9999999999999997e-37 < x

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in x around inf 91.6%

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

   if -3.5999999999999998e-22 < x < 4.9999999999999997e-37

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in z around inf 72.4%

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

      1. +-commutative 72.4%

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

      2. +-commutative 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in y around inf 54.2%

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

      1. +-commutative 54.2%

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

   10. Simplified 54.2%

       \[\leadsto \color{blue}{\frac{y}{t \cdot \left(x + 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9.5e-19) 1.0 (if (<= x 1.15e-35) (/ y t) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9.5e-19) {
		tmp = 1.0;
	} else if (x <= 1.15e-35) {
		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 <= (-9.5d-19)) then
        tmp = 1.0d0
    else if (x <= 1.15d-35) 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 <= -9.5e-19) {
		tmp = 1.0;
	} else if (x <= 1.15e-35) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -9.5e-19:
		tmp = 1.0
	elif x <= 1.15e-35:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9.5e-19)
		tmp = 1.0;
	elseif (x <= 1.15e-35)
		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 <= -9.5e-19)
		tmp = 1.0;
	elseif (x <= 1.15e-35)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9.5e-19], 1.0, If[LessEqual[x, 1.15e-35], N[(y / t), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999995e-19 or 1.1499999999999999e-35 < x

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in x around inf 91.6%

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

   if -9.4999999999999995e-19 < x < 1.1499999999999999e-35

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in x around 0 54.2%

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

Alternative 11: 53.7% 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 88.1%

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

   1. *-commutative 88.1%

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

3. Simplified 88.1%

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

5. Taylor expanded in x around inf 60.5%

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

Developer Target 1: 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 2024150 
(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)))