Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Percentage Accurate: 85.4% → 99.8%

Time: 20.6s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

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 * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

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

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log((1.0 - y)))) - t

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

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

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log((1.0 - y)))) - t

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma z (log1p (- y)) (- (* x (log y)) t)))

C

double code(double x, double y, double z, double t) {
	return fma(z, log1p(-y), ((x * log(y)) - t));
}

Julia

function code(x, y, z, t)
	return fma(z, log1p(Float64(-y)), Float64(Float64(x * log(y)) - t))
end

Wolfram

code[x_, y_, z_, t_] := N[(z * N[Log[1 + (-y)], $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Step-by-step derivation

   1. +-commutative 83.1%

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

   2. associate--l+ 83.1%

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

   3. fma-define 83.1%

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

   4. sub-neg 83.1%

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

   5. log1p-define 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-184}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-204}:\\ \;\;\;\;t \cdot \left(x \cdot \frac{\log y}{t} + -1\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-146}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7.5e-41)
   (fma x (log y) (- t))
   (if (<= x -1.4e-184)
     (- (* y (* z (+ (* y -0.5) -1.0))) t)
     (if (<= x -2.6e-204)
       (* t (+ (* x (/ (log y) t)) -1.0))
       (if (<= x 3.7e-146)
         (- (* z (log1p (- y))) t)
         (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.5e-41) {
		tmp = fma(x, log(y), -t);
	} else if (x <= -1.4e-184) {
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
	} else if (x <= -2.6e-204) {
		tmp = t * ((x * (log(y) / t)) + -1.0);
	} else if (x <= 3.7e-146) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = ((x * log(y)) + (z * log((1.0 - y)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7.5e-41)
		tmp = fma(x, log(y), Float64(-t));
	elseif (x <= -1.4e-184)
		tmp = Float64(Float64(y * Float64(z * Float64(Float64(y * -0.5) + -1.0))) - t);
	elseif (x <= -2.6e-204)
		tmp = Float64(t * Float64(Float64(x * Float64(log(y) / t)) + -1.0));
	elseif (x <= 3.7e-146)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(Float64(x * log(y)) + Float64(z * log(Float64(1.0 - y)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7.5e-41], N[(x * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[x, -1.4e-184], N[(N[(y * N[(z * N[(N[(y * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, -2.6e-204], N[(t * N[(N[(x * N[(N[Log[y], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e-146], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.50000000000000049e-41

   1. Initial program 98.3%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.3%

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

      1. fma-neg 98.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -t\right)} \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -t\right)} \]

   if -7.50000000000000049e-41 < x < -1.3999999999999999e-184

   1. Initial program 62.7%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 49.9%

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

      1. sub-neg 49.9%

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

      2. log1p-undefine 87.2%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   5. Simplified 87.2%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

   6. Taylor expanded in y around 0 87.2%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in z around 0 87.2%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

   8. Taylor expanded in y around 0 87.2%

      \[\leadsto y \cdot \left(z \cdot \left(\color{blue}{-0.5 \cdot y} - 1\right)\right) - t \]
   9. Step-by-step derivation

      1. *-commutative 87.2%

         \[\leadsto y \cdot \left(z \cdot \left(\color{blue}{y \cdot -0.5} - 1\right)\right) - t \]

   10. Simplified 87.2%

       \[\leadsto y \cdot \left(z \cdot \left(\color{blue}{y \cdot -0.5} - 1\right)\right) - t \]

   if -1.3999999999999999e-184 < x < -2.59999999999999983e-204

   1. Initial program 86.3%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 86.3%

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

      1. fma-neg 86.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -t\right)} \]

   5. Simplified 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -t\right)} \]

   6. Taylor expanded in t around inf 86.3%

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

      1. sub-neg 86.3%

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

      2. associate-/l* 86.5%

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

      3. metadata-eval 86.5%

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

   8. Simplified 86.5%

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

   if -2.59999999999999983e-204 < x < 3.69999999999999986e-146

   1. Initial program 60.1%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 52.3%

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

      1. sub-neg 52.3%

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

      2. log1p-undefine 92.3%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   5. Simplified 92.3%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

   if 3.69999999999999986e-146 < x

   1. Initial program 90.2%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing
3. Recombined 5 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-184}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-204}:\\ \;\;\;\;t \cdot \left(x \cdot \frac{\log y}{t} + -1\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-146}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(-y\right)\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+70} \lor \neg \left(t \leq 1.5 \cdot 10^{-161}\right):\\ \;\;\;\;t \cdot \left(x \cdot \frac{\log y}{t} + \left(z \cdot \frac{t\_1}{t} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log y + \frac{z \cdot t\_1 - t}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (log1p (- y))))
   (if (or (<= t -2.1e+70) (not (<= t 1.5e-161)))
     (* t (+ (* x (/ (log y) t)) (+ (* z (/ t_1 t)) -1.0)))
     (* x (+ (log y) (/ (- (* z t_1) t) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log1p(-y);
	double tmp;
	if ((t <= -2.1e+70) || !(t <= 1.5e-161)) {
		tmp = t * ((x * (log(y) / t)) + ((z * (t_1 / t)) + -1.0));
	} else {
		tmp = x * (log(y) + (((z * t_1) - t) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log1p(-y);
	double tmp;
	if ((t <= -2.1e+70) || !(t <= 1.5e-161)) {
		tmp = t * ((x * (Math.log(y) / t)) + ((z * (t_1 / t)) + -1.0));
	} else {
		tmp = x * (Math.log(y) + (((z * t_1) - t) / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log1p(-y)
	tmp = 0
	if (t <= -2.1e+70) or not (t <= 1.5e-161):
		tmp = t * ((x * (math.log(y) / t)) + ((z * (t_1 / t)) + -1.0))
	else:
		tmp = x * (math.log(y) + (((z * t_1) - t) / x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = log1p(Float64(-y))
	tmp = 0.0
	if ((t <= -2.1e+70) || !(t <= 1.5e-161))
		tmp = Float64(t * Float64(Float64(x * Float64(log(y) / t)) + Float64(Float64(z * Float64(t_1 / t)) + -1.0)));
	else
		tmp = Float64(x * Float64(log(y) + Float64(Float64(Float64(z * t_1) - t) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Log[1 + (-y)], $MachinePrecision]}, If[Or[LessEqual[t, -2.1e+70], N[Not[LessEqual[t, 1.5e-161]], $MachinePrecision]], N[(t * N[(N[(x * N[(N[Log[y], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Log[y], $MachinePrecision] + N[(N[(N[(z * t$95$1), $MachinePrecision] - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.10000000000000008e70 or 1.49999999999999994e-161 < t

   1. Initial program 90.1%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 88.8%

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

      1. associate--l+ 88.8%

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

      2. associate-/l* 88.8%

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

      3. associate-/l* 88.8%

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

      4. sub-neg 88.8%

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

      5. log1p-undefine 98.5%

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

   5. Simplified 98.5%

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

   if -2.10000000000000008e70 < t < 1.49999999999999994e-161

   1. Initial program 72.4%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 72.0%

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

      1. associate--l+ 72.0%

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

      2. div-sub 72.0%

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

      3. sub-neg 72.0%

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

      4. log1p-undefine 94.0%

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

   5. Simplified 94.0%

      \[\leadsto \color{blue}{x \cdot \left(\log y + \frac{z \cdot \mathsf{log1p}\left(-y\right) - t}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+70} \lor \neg \left(t \leq 1.5 \cdot 10^{-161}\right):\\ \;\;\;\;t \cdot \left(x \cdot \frac{\log y}{t} + \left(z \cdot \frac{\mathsf{log1p}\left(-y\right)}{t} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log y + \frac{z \cdot \mathsf{log1p}\left(-y\right) - t}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-167} \lor \neg \left(x \leq 6 \cdot 10^{-138}\right):\\ \;\;\;\;x \cdot \left(\log y + \frac{t\_1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z (log1p (- y))) t)))
   (if (or (<= x -3.1e-167) (not (<= x 6e-138)))
     (* x (+ (log y) (/ t_1 x)))
     t_1)))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * log1p(-y)) - t;
	double tmp;
	if ((x <= -3.1e-167) || !(x <= 6e-138)) {
		tmp = x * (log(y) + (t_1 / x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * Math.log1p(-y)) - t;
	double tmp;
	if ((x <= -3.1e-167) || !(x <= 6e-138)) {
		tmp = x * (Math.log(y) + (t_1 / x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * math.log1p(-y)) - t
	tmp = 0
	if (x <= -3.1e-167) or not (x <= 6e-138):
		tmp = x * (math.log(y) + (t_1 / x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * log1p(Float64(-y))) - t)
	tmp = 0.0
	if ((x <= -3.1e-167) || !(x <= 6e-138))
		tmp = Float64(x * Float64(log(y) + Float64(t_1 / x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[Or[LessEqual[x, -3.1e-167], N[Not[LessEqual[x, 6e-138]], $MachinePrecision]], N[(x * N[(N[Log[y], $MachinePrecision] + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e-167 or 6.0000000000000001e-138 < x

   1. Initial program 89.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 88.9%

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

      1. associate--l+ 88.9%

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

      2. div-sub 88.9%

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

      3. sub-neg 88.9%

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

      4. log1p-undefine 98.3%

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

   5. Simplified 98.3%

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

   if -3.1e-167 < x < 6.0000000000000001e-138

   1. Initial program 62.0%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 53.1%

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

      1. sub-neg 53.1%

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

      2. log1p-undefine 91.1%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   5. Simplified 91.1%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-167} \lor \neg \left(x \leq 6 \cdot 10^{-138}\right):\\ \;\;\;\;x \cdot \left(\log y + \frac{z \cdot \mathsf{log1p}\left(-y\right) - t}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-40} \lor \neg \left(x \leq 5.6 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.5e-40) (not (<= x 5.6e-9)))
   (fma x (log y) (- t))
   (-
    (* y (* z (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.5e-40) || !(x <= 5.6e-9)) {
		tmp = fma(x, log(y), -t);
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.5e-40) || !(x <= 5.6e-9))
		tmp = fma(x, log(y), Float64(-t));
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.5e-40], N[Not[LessEqual[x, 5.6e-9]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5000000000000001e-40 or 5.59999999999999969e-9 < x

   1. Initial program 97.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 97.5%

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

      1. fma-neg 97.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -t\right)} \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -t\right)} \]

   if -1.5000000000000001e-40 < x < 5.59999999999999969e-9

   1. Initial program 67.1%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.9%

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

      1. sub-neg 54.9%

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

      2. log1p-undefine 87.8%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

   6. Taylor expanded in y around 0 87.8%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in z around 0 87.8%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-40} \lor \neg \left(x \leq 5.6 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-78}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-79} \lor \neg \left(x \leq 2.45 \cdot 10^{+61}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -200000.0)
     t_1
     (if (<= x -2.7e-78)
       (- (- t) (* z y))
       (if (or (<= x -1.65e-79) (not (<= x 2.45e+61)))
         t_1
         (-
          (*
           y
           (*
            z
            (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
          t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -200000.0) {
		tmp = t_1;
	} else if (x <= -2.7e-78) {
		tmp = -t - (z * y);
	} else if ((x <= -1.65e-79) || !(x <= 2.45e+61)) {
		tmp = t_1;
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - 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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-200000.0d0)) then
        tmp = t_1
    else if (x <= (-2.7d-78)) then
        tmp = -t - (z * y)
    else if ((x <= (-1.65d-79)) .or. (.not. (x <= 2.45d+61))) then
        tmp = t_1
    else
        tmp = (y * (z * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -200000.0) {
		tmp = t_1;
	} else if (x <= -2.7e-78) {
		tmp = -t - (z * y);
	} else if ((x <= -1.65e-79) || !(x <= 2.45e+61)) {
		tmp = t_1;
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -200000.0:
		tmp = t_1
	elif x <= -2.7e-78:
		tmp = -t - (z * y)
	elif (x <= -1.65e-79) or not (x <= 2.45e+61):
		tmp = t_1
	else:
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -200000.0)
		tmp = t_1;
	elseif (x <= -2.7e-78)
		tmp = Float64(Float64(-t) - Float64(z * y));
	elseif ((x <= -1.65e-79) || !(x <= 2.45e+61))
		tmp = t_1;
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -200000.0)
		tmp = t_1;
	elseif (x <= -2.7e-78)
		tmp = -t - (z * y);
	elseif ((x <= -1.65e-79) || ~((x <= 2.45e+61)))
		tmp = t_1;
	else
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -200000.0], t$95$1, If[LessEqual[x, -2.7e-78], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.65e-79], N[Not[LessEqual[x, 2.45e+61]], $MachinePrecision]], t$95$1, N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e5 or -2.69999999999999994e-78 < x < -1.6499999999999999e-79 or 2.45000000000000013e61 < x

   1. Initial program 98.1%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.1%

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

      1. fma-neg 98.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -t\right)} \]

   5. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, -t\right)} \]

   6. Taylor expanded in x around inf 81.0%

      \[\leadsto \color{blue}{x \cdot \log y} \]

   if -2e5 < x < -2.69999999999999994e-78

   1. Initial program 84.6%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.3%

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

      1. sub-neg 71.3%

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

      2. log1p-undefine 86.8%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   5. Simplified 86.8%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

   6. Taylor expanded in y around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. distribute-rgt-neg-in 86.8%

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

   8. Simplified 86.8%

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

   if -1.6499999999999999e-79 < x < 2.45000000000000013e61

   1. Initial program 67.3%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 56.3%

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

      1. sub-neg 56.3%

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

      2. log1p-undefine 88.9%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   5. Simplified 88.9%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

   6. Taylor expanded in y around 0 88.9%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in z around 0 88.9%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-78}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-79} \lor \neg \left(x \leq 2.45 \cdot 10^{+61}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-40} \lor \neg \left(x \leq 3.7 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.2e-40) (not (<= x 3.7e-7)))
   (- (* x (log y)) t)
   (-
    (* y (* z (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.2e-40) || !(x <= 3.7e-7)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - 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 <= (-1.2d-40)) .or. (.not. (x <= 3.7d-7))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y * (z * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
    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-40) || !(x <= 3.7e-7)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.2e-40) or not (x <= 3.7e-7):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.2e-40) || !(x <= 3.7e-7))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.2e-40) || ~((x <= 3.7e-7)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.2e-40], N[Not[LessEqual[x, 3.7e-7]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996e-40 or 3.70000000000000004e-7 < x

   1. Initial program 97.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 97.5%

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

   if -1.19999999999999996e-40 < x < 3.70000000000000004e-7

   1. Initial program 67.1%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.9%

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

      1. sub-neg 54.9%

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

      2. log1p-undefine 87.8%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

   6. Taylor expanded in y around 0 87.8%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in z around 0 87.8%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-40} \lor \neg \left(x \leq 3.7 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.9% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (* y (* z (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
  t))

C

double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
}

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 = (y * (z * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
}

Python

def code(x, y, z, t):
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 83.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0 39.3%

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

   1. sub-neg 39.3%

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

   2. log1p-undefine 56.0%

      \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

5. Simplified 56.0%

   \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

6. Taylor expanded in y around 0 56.0%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in z around 0 56.0%

   \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

8. Final simplification 56.0%

   \[\leadsto y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t \]
9. Add Preprocessing

Alternative 9: 57.9% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* y (* z (+ -1.0 (* y (- (* y -0.3333333333333333) 0.5))))) t))

C

double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
}

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 = (y * (z * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
}

Python

def code(x, y, z, t):
	return (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5))))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 83.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0 39.3%

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

   1. sub-neg 39.3%

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

   2. log1p-undefine 56.0%

      \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

5. Simplified 56.0%

   \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

6. Taylor expanded in y around 0 56.0%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in z around 0 56.0%

   \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

8. Taylor expanded in y around 0 56.0%

   \[\leadsto y \cdot \left(z \cdot \left(y \cdot \left(\color{blue}{-0.3333333333333333 \cdot y} - 0.5\right) - 1\right)\right) - t \]
9. Step-by-step derivation

   1. *-commutative 56.0%

      \[\leadsto y \cdot \left(z \cdot \left(y \cdot \left(\color{blue}{y \cdot -0.3333333333333333} - 0.5\right) - 1\right)\right) - t \]

10. Simplified 56.0%

    \[\leadsto y \cdot \left(z \cdot \left(y \cdot \left(\color{blue}{y \cdot -0.3333333333333333} - 0.5\right) - 1\right)\right) - t \]

11. Final simplification 56.0%

    \[\leadsto y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t \]
12. Add Preprocessing

Alternative 10: 48.9% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-38} \lor \neg \left(t \leq 1.45 \cdot 10^{-126}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.75e-38) (not (<= t 1.45e-126))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.75e-38) || !(t <= 1.45e-126)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	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 <= (-1.75d-38)) .or. (.not. (t <= 1.45d-126))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.75e-38) || !(t <= 1.45e-126)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.75e-38) or not (t <= 1.45e-126):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.75e-38) || !(t <= 1.45e-126))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.75e-38) || ~((t <= 1.45e-126)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.75e-38], N[Not[LessEqual[t, 1.45e-126]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7500000000000001e-38 or 1.44999999999999994e-126 < t

   1. Initial program 90.6%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 54.2%

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

      1. neg-mul-1 54.2%

         \[\leadsto \color{blue}{-t} \]

   5. Simplified 54.2%

      \[\leadsto \color{blue}{-t} \]

   if -1.7500000000000001e-38 < t < 1.44999999999999994e-126

   1. Initial program 69.3%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 10.8%

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

      1. sub-neg 10.8%

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

      2. log1p-undefine 41.2%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   5. Simplified 41.2%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

   6. Taylor expanded in y around 0 41.2%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in y around 0 40.3%

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

      1. associate-*r* 40.3%

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

      2. neg-mul-1 40.3%

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

      3. *-commutative 40.3%

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

   9. Simplified 40.3%

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

   10. Taylor expanded in z around inf 33.5%

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

       1. mul-1-neg 33.5%

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

       2. distribute-rgt-neg-in 33.5%

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

   12. Simplified 33.5%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-38} \lor \neg \left(t \leq 1.45 \cdot 10^{-126}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* y (* z (+ (* y -0.5) -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (y * (z * ((y * -0.5) + -1.0))) - t;
}

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 = (y * (z * ((y * (-0.5d0)) + (-1.0d0)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * (z * ((y * -0.5) + -1.0))) - t;
}

Python

def code(x, y, z, t):
	return (y * (z * ((y * -0.5) + -1.0))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(z * Float64(Float64(y * -0.5) + -1.0))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(z * N[(N[(y * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 83.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0 39.3%

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

   1. sub-neg 39.3%

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

   2. log1p-undefine 56.0%

      \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

5. Simplified 56.0%

   \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

6. Taylor expanded in y around 0 56.0%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in z around 0 56.0%

   \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

8. Taylor expanded in y around 0 55.9%

   \[\leadsto y \cdot \left(z \cdot \left(\color{blue}{-0.5 \cdot y} - 1\right)\right) - t \]
9. Step-by-step derivation

   1. *-commutative 55.9%

      \[\leadsto y \cdot \left(z \cdot \left(\color{blue}{y \cdot -0.5} - 1\right)\right) - t \]

10. Simplified 55.9%

    \[\leadsto y \cdot \left(z \cdot \left(\color{blue}{y \cdot -0.5} - 1\right)\right) - t \]

11. Final simplification 55.9%

    \[\leadsto y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t \]
12. Add Preprocessing

Alternative 12: 57.5% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

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 = -t - (z * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0 39.3%

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

   1. sub-neg 39.3%

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

   2. log1p-undefine 56.0%

      \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

5. Simplified 56.0%

   \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right) - t} \]

6. Taylor expanded in y around 0 55.4%

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

   1. mul-1-neg 55.4%

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

   2. distribute-rgt-neg-in 55.4%

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

8. Simplified 55.4%

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

9. Final simplification 55.4%

   \[\leadsto \left(-t\right) - z \cdot y \]
10. Add Preprocessing

Alternative 13: 43.1% accurate, 105.5× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

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

C

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

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 = -t
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return Float64(-t)
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 83.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in t around inf 38.3%

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

   1. neg-mul-1 38.3%

      \[\leadsto \color{blue}{-t} \]

5. Simplified 38.3%

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

Alternative 14: 2.2% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

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 = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in z around inf 64.5%

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

4. Taylor expanded in t around inf 30.1%

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

   1. associate-*r/ 30.1%

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

   2. mul-1-neg 30.1%

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

6. Simplified 30.1%

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

   1. expm1-log1p-u 16.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(z \cdot \frac{-t}{z}\right)\right)} \]

   2. expm1-undefine 12.2%

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

   3. distribute-frac-neg 12.2%

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

8. Applied egg-rr 12.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(z \cdot \left(-\frac{t}{z}\right)\right)} - 1} \]
9. Step-by-step derivation

   1. sub-neg 12.2%

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

   2. metadata-eval 12.2%

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

   3. +-commutative 12.2%

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

   4. log1p-undefine 12.2%

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

   5. rem-exp-log 25.9%

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

   6. distribute-neg-frac 25.9%

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

   7. associate-/l* 28.2%

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

   8. *-commutative 28.2%

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

   9. associate-/l* 34.1%

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

   10. *-inverses 34.1%

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

   11. *-rgt-identity 34.1%

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

   12. unsub-neg 34.1%

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

10. Simplified 34.1%

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

    1. associate-+r- 38.3%

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

    2. metadata-eval 38.3%

       \[\leadsto \color{blue}{0} - t \]

    3. neg-sub0 38.3%

       \[\leadsto \color{blue}{-t} \]

    4. add-sqr-sqrt 16.7%

       \[\leadsto \color{blue}{\sqrt{-t} \cdot \sqrt{-t}} \]

    5. sqrt-unprod 11.7%

       \[\leadsto \color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}} \]

    6. sqr-neg 11.7%

       \[\leadsto \sqrt{\color{blue}{t \cdot t}} \]

    7. sqrt-unprod 1.3%

       \[\leadsto \color{blue}{\sqrt{t} \cdot \sqrt{t}} \]

    8. add-sqr-sqrt 2.3%

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

    9. *-un-lft-identity 2.3%

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

12. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{1 \cdot t} \]
13. Step-by-step derivation

    1. *-lft-identity 2.3%

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

14. Simplified 2.3%

    \[\leadsto \color{blue}{t} \]
15. Add Preprocessing

Developer target: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (*
   (- z)
   (+
    (+ (* 0.5 (* y y)) y)
    (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y)))))
  (- t (* x (log y)))))

C

double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
}

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 = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * Math.log(y)));
}

Python

def code(x, y, z, t):
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * math.log(y)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-z) * Float64(Float64(Float64(0.5 * Float64(y * y)) + y) + Float64(Float64(0.3333333333333333 / Float64(1.0 * Float64(1.0 * 1.0))) * Float64(y * Float64(y * y))))) - Float64(t - Float64(x * log(y))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-z) * N[(N[(N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(1.0 * N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024107 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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