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

Percentage Accurate: 89.1% → 96.9%

Time: 10.3s

Alternatives: 12

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: 89.1% 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: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{y \cdot z - x}{t_2}}{x + 1}\\ t_4 := \frac{y}{x + 1}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_4 \cdot \frac{z}{t_2}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\frac{t_4}{t} - \frac{t_1}{z \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0)))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ (- (* y z) x) t_2)) (+ x 1.0)))
        (t_4 (/ y (+ x 1.0))))
   (if (<= t_3 (- INFINITY))
     (* t_4 (/ z t_2))
     (if (<= t_3 5e+276) t_3 (+ t_1 (- (/ t_4 t) (/ t_1 (* z t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = y / (x + 1.0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4 * (z / t_2);
	} else if (t_3 <= 5e+276) {
		tmp = t_3;
	} else {
		tmp = t_1 + ((t_4 / t) - (t_1 / (z * t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double t_2 = (z * t) - x;
	double t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	double t_4 = y / (x + 1.0);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_4 * (z / t_2);
	} else if (t_3 <= 5e+276) {
		tmp = t_3;
	} else {
		tmp = t_1 + ((t_4 / t) - (t_1 / (z * t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x + 1.0)
	t_2 = (z * t) - x
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0)
	t_4 = y / (x + 1.0)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_4 * (z / t_2)
	elif t_3 <= 5e+276:
		tmp = t_3
	else:
		tmp = t_1 + ((t_4 / t) - (t_1 / (z * t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_2)) / Float64(x + 1.0))
	t_4 = Float64(y / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(t_4 * Float64(z / t_2));
	elseif (t_3 <= 5e+276)
		tmp = t_3;
	else
		tmp = Float64(t_1 + Float64(Float64(t_4 / t) - Float64(t_1 / Float64(z * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x + 1.0);
	t_2 = (z * t) - x;
	t_3 = (x + (((y * z) - x) / t_2)) / (x + 1.0);
	t_4 = y / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_4 * (z / t_2);
	elseif (t_3 <= 5e+276)
		tmp = t_3;
	else
		tmp = t_1 + ((t_4 / t) - (t_1 / (z * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(t$95$4 * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+276], t$95$3, N[(t$95$1 + N[(N[(t$95$4 / t), $MachinePrecision] - N[(t$95$1 / N[(z * t), $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 36.7%

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

      1. *-commutative 36.7%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in y around inf 35.8%

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

      1. *-commutative 35.8%

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

      2. *-commutative 35.8%

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

      3. times-frac 83.4%

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

      4. +-commutative 83.4%

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

      5. *-commutative 83.4%

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

   6. Simplified 83.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 5.00000000000000001e276

   1. Initial program 99.4%

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

   if 5.00000000000000001e276 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 19.8%

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

      1. *-commutative 19.8%

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

   3. Simplified 19.8%

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

      1. clear-num 19.8%

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

      2. inv-pow 19.8%

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

   5. Applied egg-rr 19.8%

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

      1. unpow-1 19.8%

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

      2. *-commutative 19.8%

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

      3. *-commutative 19.8%

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

   7. Simplified 19.8%

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

   8. Taylor expanded in t around inf 81.8%

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

      1. associate--l+ 81.8%

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

      2. +-commutative 81.8%

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

      3. associate-/r* 81.9%

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

      4. +-commutative 81.9%

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

      5. associate-/r* 81.9%

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

      6. +-commutative 81.9%

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

      7. *-commutative 81.9%

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

   10. Simplified 81.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{z \cdot t - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(\frac{\frac{y}{x + 1}}{t} - \frac{\frac{x}{x + 1}}{z \cdot t}\right)\\ \end{array} \]

Alternative 2: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{t_1}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

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 (- INFINITY))
     (* (/ y (+ x 1.0)) (/ z t_1))
     (if (<= t_2 5e+276) 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 <= -((double) INFINITY)) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 5e+276) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 5e+276) {
		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 <= -math.inf:
		tmp = (y / (x + 1.0)) * (z / t_1)
	elif t_2 <= 5e+276:
		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 <= Float64(-Inf))
		tmp = Float64(Float64(y / Float64(x + 1.0)) * Float64(z / t_1));
	elseif (t_2 <= 5e+276)
		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 <= -Inf)
		tmp = (y / (x + 1.0)) * (z / t_1);
	elseif (t_2 <= 5e+276)
		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, (-Infinity)], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+276], 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)) < -inf.0

   1. Initial program 36.7%

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

      1. *-commutative 36.7%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in y around inf 35.8%

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

      1. *-commutative 35.8%

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

      2. *-commutative 35.8%

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

      3. times-frac 83.4%

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

      4. +-commutative 83.4%

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

      5. *-commutative 83.4%

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

   6. Simplified 83.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 5.00000000000000001e276

   1. Initial program 99.4%

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

   if 5.00000000000000001e276 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 19.8%

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

      1. *-commutative 19.8%

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

   3. Simplified 19.8%

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

   4. Taylor expanded in z around inf 81.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{z \cdot t - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 3: 83.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-39} \lor \neg \left(t \leq 5.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x + 1} \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.2e-39) (not (<= t 5.5e-35)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (- 1.0 (* (/ y (+ x 1.0)) (/ z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.2e-39) || !(t <= 5.5e-35)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - ((y / (x + 1.0)) * (z / 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 <= (-2.2d-39)) .or. (.not. (t <= 5.5d-35))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 - ((y / (x + 1.0d0)) * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.2e-39) || !(t <= 5.5e-35)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - ((y / (x + 1.0)) * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.2e-39) or not (t <= 5.5e-35):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 - ((y / (x + 1.0)) * (z / x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.2e-39) || !(t <= 5.5e-35))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 - Float64(Float64(y / Float64(x + 1.0)) * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.2e-39) || ~((t <= 5.5e-35)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 - ((y / (x + 1.0)) * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.2e-39], N[Not[LessEqual[t, 5.5e-35]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.20000000000000001e-39 or 5.4999999999999997e-35 < t

   1. Initial program 87.4%

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

      1. *-commutative 87.4%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in z around inf 86.4%

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

   if -2.20000000000000001e-39 < t < 5.4999999999999997e-35

   1. Initial program 92.6%

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

      1. *-commutative 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in t around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. +-commutative 78.9%

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

      3. associate-+l+ 78.9%

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

      4. +-commutative 78.9%

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

      5. mul-1-neg 78.9%

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

      6. unsub-neg 78.9%

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

      7. associate-/l* 81.8%

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

      8. +-commutative 81.8%

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

   6. Simplified 81.8%

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

   7. Taylor expanded in y around 0 78.7%

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

      1. mul-1-neg 78.7%

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

      2. unsub-neg 78.7%

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

      3. times-frac 84.4%

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

      4. +-commutative 84.4%

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

   9. Simplified 84.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-39} \lor \neg \left(t \leq 5.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x + 1} \cdot \frac{z}{x}\\ \end{array} \]

Alternative 4: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-24} \lor \neg \left(x \leq 1.45 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot t - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.22e-24) (not (<= x 1.45e-63)))
   (/ x (+ x 1.0))
   (* y (/ z (- (* z t) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.22e-24) || !(x <= 1.45e-63)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = y * (z / ((z * t) - 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 <= (-1.22d-24)) .or. (.not. (x <= 1.45d-63))) then
        tmp = x / (x + 1.0d0)
    else
        tmp = y * (z / ((z * t) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.22e-24) || !(x <= 1.45e-63)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = y * (z / ((z * t) - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.22e-24) or not (x <= 1.45e-63):
		tmp = x / (x + 1.0)
	else:
		tmp = y * (z / ((z * t) - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.22e-24) || !(x <= 1.45e-63))
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(y * Float64(z / Float64(Float64(z * t) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.22e-24) || ~((x <= 1.45e-63)))
		tmp = x / (x + 1.0);
	else
		tmp = y * (z / ((z * t) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.22e-24], N[Not[LessEqual[x, 1.45e-63]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.22000000000000004e-24 or 1.44999999999999987e-63 < x

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in t around inf 83.8%

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

      1. +-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -1.22000000000000004e-24 < x < 1.44999999999999987e-63

   1. Initial program 92.0%

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

      1. *-commutative 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in y around inf 56.8%

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

      1. *-commutative 56.8%

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

      2. *-commutative 56.8%

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

      3. times-frac 60.6%

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

      4. +-commutative 60.6%

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

      5. *-commutative 60.6%

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

   6. Simplified 60.6%

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

   7. Taylor expanded in x around 0 60.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-24} \lor \neg \left(x \leq 1.45 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot t - x}\\ \end{array} \]

Alternative 5: 78.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-137} \lor \neg \left(t \leq 3.7 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.8e-137) (not (<= t 3.7e-63)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (- 1.0 (* y (/ z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.8e-137) || !(t <= 3.7e-63)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - (y * (z / 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 <= (-4.8d-137)) .or. (.not. (t <= 3.7d-63))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 - (y * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.8e-137) || !(t <= 3.7e-63)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - (y * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.8e-137) or not (t <= 3.7e-63):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 - (y * (z / x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.8e-137) || !(t <= 3.7e-63))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 - Float64(y * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.8e-137) || ~((t <= 3.7e-63)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 - (y * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.8e-137], N[Not[LessEqual[t, 3.7e-63]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8000000000000001e-137 or 3.70000000000000012e-63 < t

   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. Taylor expanded in z around inf 81.9%

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

   if -4.8000000000000001e-137 < t < 3.70000000000000012e-63

   1. Initial program 92.8%

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

      1. *-commutative 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in t around 0 83.2%

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

      1. +-commutative 83.2%

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

      2. +-commutative 83.2%

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

      3. associate-+l+ 83.2%

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

      4. +-commutative 83.2%

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

      5. mul-1-neg 83.2%

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

      6. unsub-neg 83.2%

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

      7. associate-/l* 86.9%

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

      8. +-commutative 86.9%

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

   6. Simplified 86.9%

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

   7. Taylor expanded in y around 0 83.0%

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

      1. mul-1-neg 83.0%

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

      2. unsub-neg 83.0%

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

      3. times-frac 89.2%

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

      4. +-commutative 89.2%

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

   9. Simplified 89.2%

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

   10. Taylor expanded in x around 0 79.4%

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

       1. *-commutative 79.4%

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

       2. associate-*l/ 79.5%

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

       3. *-commutative 79.5%

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

   12. Simplified 79.5%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-137} \lor \neg \left(t \leq 3.7 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \frac{z}{x}\\ \end{array} \]

Alternative 6: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-23}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot t - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.05e-23)
   (- 1.0 (* (/ z x) (/ y x)))
   (if (<= x 3.3e-63) (* y (/ z (- (* z t) x))) (/ x (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e-23) {
		tmp = 1.0 - ((z / x) * (y / x));
	} else if (x <= 3.3e-63) {
		tmp = y * (z / ((z * t) - x));
	} else {
		tmp = 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 (x <= (-1.05d-23)) then
        tmp = 1.0d0 - ((z / x) * (y / x))
    else if (x <= 3.3d-63) then
        tmp = y * (z / ((z * t) - x))
    else
        tmp = 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 (x <= -1.05e-23) {
		tmp = 1.0 - ((z / x) * (y / x));
	} else if (x <= 3.3e-63) {
		tmp = y * (z / ((z * t) - x));
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.05e-23:
		tmp = 1.0 - ((z / x) * (y / x))
	elif x <= 3.3e-63:
		tmp = y * (z / ((z * t) - x))
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.05e-23)
		tmp = Float64(1.0 - Float64(Float64(z / x) * Float64(y / x)));
	elseif (x <= 3.3e-63)
		tmp = Float64(y * Float64(z / Float64(Float64(z * t) - x)));
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.05e-23)
		tmp = 1.0 - ((z / x) * (y / x));
	elseif (x <= 3.3e-63)
		tmp = y * (z / ((z * t) - x));
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.05e-23], N[(1.0 - N[(N[(z / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-63], N[(y * N[(z / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e-23

   1. Initial program 91.4%

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

      1. *-commutative 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in x around -inf 70.6%

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

      1. mul-1-neg 70.6%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z - t \cdot z}{{x}^{2}}\right)} \]

      2. unsub-neg 70.6%

         \[\leadsto \color{blue}{1 - \frac{y \cdot z - t \cdot z}{{x}^{2}}} \]

      3. distribute-rgt-out-- 70.6%

         \[\leadsto 1 - \frac{\color{blue}{z \cdot \left(y - t\right)}}{{x}^{2}} \]

      4. unpow2 70.6%

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

      5. times-frac 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in y around inf 91.1%

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

   if -1.05e-23 < x < 3.29999999999999994e-63

   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 y around inf 56.2%

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

      1. *-commutative 56.2%

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

      2. *-commutative 56.2%

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

      3. times-frac 60.1%

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

      4. +-commutative 60.1%

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

      5. *-commutative 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in x around 0 60.1%

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

   if 3.29999999999999994e-63 < x

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in t around inf 83.0%

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

      1. +-commutative 83.0%

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

   6. Simplified 83.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-23}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot t - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 7: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-21}:\\ \;\;\;\;1 - \frac{\frac{y}{x}}{\frac{x}{z}}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot t - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.1e-21)
   (- 1.0 (/ (/ y x) (/ x z)))
   (if (<= x 2.2e-63) (* y (/ z (- (* z t) x))) (/ x (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.1e-21) {
		tmp = 1.0 - ((y / x) / (x / z));
	} else if (x <= 2.2e-63) {
		tmp = y * (z / ((z * t) - x));
	} else {
		tmp = 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 (x <= (-2.1d-21)) then
        tmp = 1.0d0 - ((y / x) / (x / z))
    else if (x <= 2.2d-63) then
        tmp = y * (z / ((z * t) - x))
    else
        tmp = 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 (x <= -2.1e-21) {
		tmp = 1.0 - ((y / x) / (x / z));
	} else if (x <= 2.2e-63) {
		tmp = y * (z / ((z * t) - x));
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.1e-21:
		tmp = 1.0 - ((y / x) / (x / z))
	elif x <= 2.2e-63:
		tmp = y * (z / ((z * t) - x))
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.1e-21)
		tmp = Float64(1.0 - Float64(Float64(y / x) / Float64(x / z)));
	elseif (x <= 2.2e-63)
		tmp = Float64(y * Float64(z / Float64(Float64(z * t) - x)));
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.1e-21)
		tmp = 1.0 - ((y / x) / (x / z));
	elseif (x <= 2.2e-63)
		tmp = y * (z / ((z * t) - x));
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.1e-21], N[(1.0 - N[(N[(y / x), $MachinePrecision] / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-63], N[(y * N[(z / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.10000000000000013e-21

   1. Initial program 91.4%

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

      1. *-commutative 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in x around -inf 70.6%

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

      1. mul-1-neg 70.6%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z - t \cdot z}{{x}^{2}}\right)} \]

      2. unsub-neg 70.6%

         \[\leadsto \color{blue}{1 - \frac{y \cdot z - t \cdot z}{{x}^{2}}} \]

      3. distribute-rgt-out-- 70.6%

         \[\leadsto 1 - \frac{\color{blue}{z \cdot \left(y - t\right)}}{{x}^{2}} \]

      4. unpow2 70.6%

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

      5. times-frac 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in y around inf 91.1%

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

      1. *-commutative 91.1%

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

      2. clear-num 91.1%

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

      3. un-div-inv 91.1%

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

   9. Applied egg-rr 91.1%

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

   if -2.10000000000000013e-21 < x < 2.2e-63

   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 y around inf 56.2%

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

      1. *-commutative 56.2%

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

      2. *-commutative 56.2%

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

      3. times-frac 60.1%

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

      4. +-commutative 60.1%

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

      5. *-commutative 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in x around 0 60.1%

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

   if 2.2e-63 < x

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in t around inf 83.0%

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

      1. +-commutative 83.0%

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

   6. Simplified 83.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-21}:\\ \;\;\;\;1 - \frac{\frac{y}{x}}{\frac{x}{z}}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot t - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 8: 69.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.1 \cdot 10^{-153}:\\ \;\;\;\;1 - y \cdot \frac{z}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0))))
   (if (<= x -5.8e-8)
     t_1
     (if (<= x -7.1e-153)
       (- 1.0 (* y (/ z x)))
       (if (<= x 4e-106) (/ y t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -5.8e-8) {
		tmp = t_1;
	} else if (x <= -7.1e-153) {
		tmp = 1.0 - (y * (z / x));
	} else if (x <= 4e-106) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = x / (x + 1.0d0)
    if (x <= (-5.8d-8)) then
        tmp = t_1
    else if (x <= (-7.1d-153)) then
        tmp = 1.0d0 - (y * (z / x))
    else if (x <= 4d-106) then
        tmp = y / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -5.8e-8) {
		tmp = t_1;
	} else if (x <= -7.1e-153) {
		tmp = 1.0 - (y * (z / x));
	} else if (x <= 4e-106) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -5.8e-8:
		tmp = t_1
	elif x <= -7.1e-153:
		tmp = 1.0 - (y * (z / x))
	elif x <= 4e-106:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -5.8e-8)
		tmp = t_1;
	elseif (x <= -7.1e-153)
		tmp = Float64(1.0 - Float64(y * Float64(z / x)));
	elseif (x <= 4e-106)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -5.8e-8)
		tmp = t_1;
	elseif (x <= -7.1e-153)
		tmp = 1.0 - (y * (z / x));
	elseif (x <= 4e-106)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-8], t$95$1, If[LessEqual[x, -7.1e-153], N[(1.0 - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-106], N[(y / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8000000000000003e-8 or 3.99999999999999976e-106 < 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. Taylor expanded in t around inf 81.1%

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

      1. +-commutative 81.1%

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

   6. Simplified 81.1%

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

   if -5.8000000000000003e-8 < x < -7.1000000000000001e-153

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 51.3%

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

      1. +-commutative 51.3%

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

      2. +-commutative 51.3%

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

      3. associate-+l+ 51.3%

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

      4. +-commutative 51.3%

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

      5. mul-1-neg 51.3%

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

      6. unsub-neg 51.3%

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

      7. associate-/l* 51.2%

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

      8. +-commutative 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in y around 0 51.3%

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

      1. mul-1-neg 51.3%

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

      2. unsub-neg 51.3%

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

      3. times-frac 51.2%

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

      4. +-commutative 51.2%

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

   9. Simplified 51.2%

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

   10. Taylor expanded in x around 0 51.3%

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

       1. *-commutative 51.3%

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

       2. associate-*l/ 51.2%

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

       3. *-commutative 51.2%

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

   12. Simplified 51.2%

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

   if -7.1000000000000001e-153 < x < 3.99999999999999976e-106

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

   3. Simplified 88.0%

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

      1. clear-num 88.0%

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

      2. inv-pow 88.0%

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

   5. Applied egg-rr 88.0%

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

      1. unpow-1 88.0%

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

      2. *-commutative 88.0%

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

      3. *-commutative 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in x around 0 59.4%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -7.1 \cdot 10^{-153}:\\ \;\;\;\;1 - y \cdot \frac{z}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 9: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-155} \lor \neg \left(x \leq 2.8 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7.2e-155) (not (<= x 2.8e-106))) (/ x (+ x 1.0)) (/ y t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7.2e-155) || !(x <= 2.8e-106)) {
		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 <= (-7.2d-155)) .or. (.not. (x <= 2.8d-106))) 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 <= -7.2e-155) || !(x <= 2.8e-106)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -7.2e-155) or not (x <= 2.8e-106):
		tmp = x / (x + 1.0)
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7.2e-155) || !(x <= 2.8e-106))
		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 <= -7.2e-155) || ~((x <= 2.8e-106)))
		tmp = x / (x + 1.0);
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7.2e-155], N[Not[LessEqual[x, 2.8e-106]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999977e-155 or 2.79999999999999988e-106 < x

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in t around inf 73.9%

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

      1. +-commutative 73.9%

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

   6. Simplified 73.9%

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

   if -7.19999999999999977e-155 < x < 2.79999999999999988e-106

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

   3. Simplified 87.7%

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

      1. clear-num 87.6%

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

      2. inv-pow 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. unpow-1 87.6%

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

      2. *-commutative 87.6%

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

      3. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in x around 0 61.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-155} \lor \neg \left(x \leq 2.8 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]

Alternative 10: 68.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.5e-19) 1.0 (if (<= x 1.6e-9) (/ y t) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.5e-19) {
		tmp = 1.0;
	} else if (x <= 1.6e-9) {
		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 <= (-5.5d-19)) then
        tmp = 1.0d0
    else if (x <= 1.6d-9) 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 <= -5.5e-19) {
		tmp = 1.0;
	} else if (x <= 1.6e-9) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.5e-19:
		tmp = 1.0
	elif x <= 1.6e-9:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.5e-19)
		tmp = 1.0;
	elseif (x <= 1.6e-9)
		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 <= -5.5e-19)
		tmp = 1.0;
	elseif (x <= 1.6e-9)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.5e-19], 1.0, If[LessEqual[x, 1.6e-9], N[(y / t), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999996e-19 or 1.60000000000000006e-9 < 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. Taylor expanded in z around inf 75.7%

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

   5. Taylor expanded in x around inf 86.1%

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

   if -5.4999999999999996e-19 < x < 1.60000000000000006e-9

   1. Initial program 90.9%

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

      1. *-commutative 90.9%

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

   3. Simplified 90.9%

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

      1. clear-num 90.7%

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

      2. inv-pow 90.7%

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

   5. Applied egg-rr 90.7%

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

      1. unpow-1 90.7%

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

      2. *-commutative 90.7%

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

      3. *-commutative 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in x around 0 45.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 56.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.2e-21) 1.0 (if (<= x 7.4e-9) x 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.2e-21) {
		tmp = 1.0;
	} else if (x <= 7.4e-9) {
		tmp = 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 <= (-3.2d-21)) then
        tmp = 1.0d0
    else if (x <= 7.4d-9) then
        tmp = 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 <= -3.2e-21) {
		tmp = 1.0;
	} else if (x <= 7.4e-9) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.2e-21:
		tmp = 1.0
	elif x <= 7.4e-9:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.2e-21)
		tmp = 1.0;
	elseif (x <= 7.4e-9)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.2e-21)
		tmp = 1.0;
	elseif (x <= 7.4e-9)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.2e-21], 1.0, If[LessEqual[x, 7.4e-9], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000002e-21 or 7.4e-9 < 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. Taylor expanded in z around inf 75.7%

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

   5. Taylor expanded in x around inf 86.1%

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

   if -3.2000000000000002e-21 < x < 7.4e-9

   1. Initial program 90.9%

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

      1. *-commutative 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in t around inf 21.7%

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

      1. +-commutative 21.7%

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

   6. Simplified 21.7%

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

   7. Taylor expanded in x around 0 21.7%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 54.0% 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 89.6%

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

   1. *-commutative 89.6%

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

3. Simplified 89.6%

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

4. Taylor expanded in z around inf 71.0%

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

5. Taylor expanded in x around inf 53.7%

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

6. Final simplification 53.7%

   \[\leadsto 1 \]

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 2023215 
(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)))