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

Percentage Accurate: 85.0% → 99.8%

Time: 22.0s

Alternatives: 13

Speedup: 1.9×

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: 85.0% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate--l+ 83.8%

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

   2. fma-def 83.8%

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

   3. fma-neg 83.8%

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

   4. sub-neg 83.8%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. +-commutative 83.8%

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

   2. fma-def 83.8%

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

   3. sub-neg 83.8%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate--l+ 83.8%

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

   2. fma-def 83.8%

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

   3. fma-neg 83.8%

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

   4. sub-neg 83.8%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 99.6%

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

   1. mul-1-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. unsub-neg 99.6%

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

   4. mul-1-neg 99.6%

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

   5. distribute-rgt-neg-in 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 89.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-8} \lor \neg \left(t \leq 8 \cdot 10^{-78}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -3.2e-8) (not (<= t 8e-78))) (- t_1 t) (- t_1 (* y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((t <= -3.2e-8) || !(t <= 8e-78)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (y * z);
	}
	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 ((t <= (-3.2d-8)) .or. (.not. (t <= 8d-78))) then
        tmp = t_1 - t
    else
        tmp = t_1 - (y * z)
    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 ((t <= -3.2e-8) || !(t <= 8e-78)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (t <= -3.2e-8) or not (t <= 8e-78):
		tmp = t_1 - t
	else:
		tmp = t_1 - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((t <= -3.2e-8) || !(t <= 8e-78))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((t <= -3.2e-8) || ~((t <= 8e-78)))
		tmp = t_1 - t;
	else
		tmp = t_1 - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -3.2e-8], N[Not[LessEqual[t, 8e-78]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2000000000000002e-8 or 7.99999999999999999e-78 < t

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. fma-def 94.4%

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

      3. sub-neg 94.4%

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

      4. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 93.6%

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

   if -3.2000000000000002e-8 < t < 7.99999999999999999e-78

   1. Initial program 70.2%

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

      1. associate--l+ 70.2%

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

      2. fma-def 70.2%

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

      3. fma-neg 70.2%

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

      4. sub-neg 70.2%

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

      5. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. unsub-neg 99.8%

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

      4. mul-1-neg 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. mul-1-neg 91.7%

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

      3. unsub-neg 91.7%

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

   9. Simplified 91.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-8} \lor \neg \left(t \leq 8 \cdot 10^{-78}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \]

Alternative 5: 89.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-8} \lor \neg \left(t \leq 8.6 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{x}{\frac{1}{\log y}} - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.3e-8) (not (<= t 8.6e-78)))
   (- (/ x (/ 1.0 (log y))) t)
   (- (* x (log y)) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.3e-8) || !(t <= 8.6e-78)) {
		tmp = (x / (1.0 / log(y))) - t;
	} else {
		tmp = (x * log(y)) - (y * z);
	}
	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 <= (-3.3d-8)) .or. (.not. (t <= 8.6d-78))) then
        tmp = (x / (1.0d0 / log(y))) - t
    else
        tmp = (x * log(y)) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.3e-8) || !(t <= 8.6e-78)) {
		tmp = (x / (1.0 / Math.log(y))) - t;
	} else {
		tmp = (x * Math.log(y)) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.3e-8) or not (t <= 8.6e-78):
		tmp = (x / (1.0 / math.log(y))) - t
	else:
		tmp = (x * math.log(y)) - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.3e-8) || !(t <= 8.6e-78))
		tmp = Float64(Float64(x / Float64(1.0 / log(y))) - t);
	else
		tmp = Float64(Float64(x * log(y)) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3.3e-8) || ~((t <= 8.6e-78)))
		tmp = (x / (1.0 / log(y))) - t;
	else
		tmp = (x * log(y)) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.3e-8], N[Not[LessEqual[t, 8.6e-78]], $MachinePrecision]], N[(N[(x / N[(1.0 / N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.29999999999999977e-8 or 8.59999999999999987e-78 < t

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. fma-def 94.4%

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

      3. sub-neg 94.4%

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

      4. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 93.6%

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

      1. add-log-exp 52.9%

         \[\leadsto \color{blue}{\log \left(e^{x \cdot \log y}\right)} - t \]

      2. *-commutative 52.9%

         \[\leadsto \log \left(e^{\color{blue}{\log y \cdot x}}\right) - t \]

      3. exp-to-pow 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. log-pow 93.6%

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

      2. /-rgt-identity 93.6%

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

      3. associate-/l* 93.6%

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

   8. Applied egg-rr 93.6%

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

   if -3.29999999999999977e-8 < t < 8.59999999999999987e-78

   1. Initial program 70.2%

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

      1. associate--l+ 70.2%

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

      2. fma-def 70.2%

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

      3. fma-neg 70.2%

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

      4. sub-neg 70.2%

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

      5. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. unsub-neg 99.8%

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

      4. mul-1-neg 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. mul-1-neg 91.7%

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

      3. unsub-neg 91.7%

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

   9. Simplified 91.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-8} \lor \neg \left(t \leq 8.6 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{x}{\frac{1}{\log y}} - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \]

Alternative 6: 89.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-8} \lor \neg \left(t \leq 1.8 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{\log y}{\frac{1}{x}} - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.2e-8) (not (<= t 1.8e-77)))
   (- (/ (log y) (/ 1.0 x)) t)
   (- (* x (log y)) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.2e-8) || !(t <= 1.8e-77)) {
		tmp = (log(y) / (1.0 / x)) - t;
	} else {
		tmp = (x * log(y)) - (y * z);
	}
	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 <= (-3.2d-8)) .or. (.not. (t <= 1.8d-77))) then
        tmp = (log(y) / (1.0d0 / x)) - t
    else
        tmp = (x * log(y)) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.2e-8) || !(t <= 1.8e-77)) {
		tmp = (Math.log(y) / (1.0 / x)) - t;
	} else {
		tmp = (x * Math.log(y)) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.2e-8) or not (t <= 1.8e-77):
		tmp = (math.log(y) / (1.0 / x)) - t
	else:
		tmp = (x * math.log(y)) - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.2e-8) || !(t <= 1.8e-77))
		tmp = Float64(Float64(log(y) / Float64(1.0 / x)) - t);
	else
		tmp = Float64(Float64(x * log(y)) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3.2e-8) || ~((t <= 1.8e-77)))
		tmp = (log(y) / (1.0 / x)) - t;
	else
		tmp = (x * log(y)) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.2e-8], N[Not[LessEqual[t, 1.8e-77]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2000000000000002e-8 or 1.8e-77 < t

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. fma-def 94.4%

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

      3. sub-neg 94.4%

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

      4. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 93.6%

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

      1. add-log-exp 52.9%

         \[\leadsto \color{blue}{\log \left(e^{x \cdot \log y}\right)} - t \]

      2. *-commutative 52.9%

         \[\leadsto \log \left(e^{\color{blue}{\log y \cdot x}}\right) - t \]

      3. exp-to-pow 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. log-pow 93.6%

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

      2. /-rgt-identity 93.6%

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

      3. *-commutative 93.6%

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

      4. associate-/l* 93.6%

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

   8. Applied egg-rr 93.6%

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

   if -3.2000000000000002e-8 < t < 1.8e-77

   1. Initial program 70.2%

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

      1. associate--l+ 70.2%

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

      2. fma-def 70.2%

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

      3. fma-neg 70.2%

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

      4. sub-neg 70.2%

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

      5. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. unsub-neg 99.8%

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

      4. mul-1-neg 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. mul-1-neg 91.7%

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

      3. unsub-neg 91.7%

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

   9. Simplified 91.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-8} \lor \neg \left(t \leq 1.8 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{\log y}{\frac{1}{x}} - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \]

Alternative 7: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-129} \lor \neg \left(x \leq 2.9 \cdot 10^{-46}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.4e-129) (not (<= x 2.9e-46)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e-129) || !(x <= 2.9e-46)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e-129) || !(x <= 2.9e-46)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.4e-129) or not (x <= 2.9e-46):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.4e-129) || !(x <= 2.9e-46))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.4e-129], N[Not[LessEqual[x, 2.9e-46]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999989e-129 or 2.90000000000000005e-46 < x

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. fma-def 93.9%

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

      3. sub-neg 93.9%

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

      4. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 93.6%

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

   if -2.39999999999999989e-129 < x < 2.90000000000000005e-46

   1. Initial program 68.9%

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

   2. Taylor expanded in x around 0 60.4%

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

      1. sub-neg 60.4%

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

      2. mul-1-neg 60.4%

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

      3. log1p-def 91.5%

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

      4. mul-1-neg 91.5%

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

   4. Simplified 91.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-129} \lor \neg \left(x \leq 2.9 \cdot 10^{-46}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 8: 89.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-134} \lor \neg \left(x \leq 6.6 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5e-134) (not (<= x 6.6e-44)))
   (- (* x (log y)) t)
   (- (- t) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e-134) || !(x <= 6.6e-44)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -t - (y * z);
	}
	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 <= (-5d-134)) .or. (.not. (x <= 6.6d-44))) then
        tmp = (x * log(y)) - t
    else
        tmp = -t - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e-134) || !(x <= 6.6e-44)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5e-134) or not (x <= 6.6e-44):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -t - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5e-134) || !(x <= 6.6e-44))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-t) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5e-134) || ~((x <= 6.6e-44)))
		tmp = (x * log(y)) - t;
	else
		tmp = -t - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5e-134], N[Not[LessEqual[x, 6.6e-44]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000003e-134 or 6.60000000000000011e-44 < x

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. fma-def 93.9%

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

      3. sub-neg 93.9%

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

      4. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 93.6%

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

   if -5.0000000000000003e-134 < x < 6.60000000000000011e-44

   1. Initial program 68.9%

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

      1. associate--l+ 68.9%

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

      2. fma-def 68.9%

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

      3. fma-neg 68.9%

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

      4. sub-neg 68.9%

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

      5. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. unsub-neg 99.5%

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

      4. mul-1-neg 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 91.1%

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

      1. associate-*r* 91.1%

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

      2. neg-mul-1 91.1%

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

   9. Simplified 91.1%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-134} \lor \neg \left(x \leq 6.6 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 9: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) - (y * z)) - 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)) - (y * z)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

2. Taylor expanded in y around 0 99.6%

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

   1. log-pow 49.8%

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

   2. +-commutative 49.8%

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

   3. mul-1-neg 49.8%

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

   4. unsub-neg 49.8%

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

   5. log-pow 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \left(x \cdot \log y - y \cdot z\right) - t \]

Alternative 10: 77.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+124} \lor \neg \left(x \leq 3.3 \cdot 10^{+105}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.8e+124) (not (<= x 3.3e+105)))
   (* x (log y))
   (- (- t) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.8e+124) || !(x <= 3.3e+105)) {
		tmp = x * log(y);
	} else {
		tmp = -t - (y * z);
	}
	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 <= (-5.8d+124)) .or. (.not. (x <= 3.3d+105))) then
        tmp = x * log(y)
    else
        tmp = -t - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.8e+124) || !(x <= 3.3e+105)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.8e+124) or not (x <= 3.3e+105):
		tmp = x * math.log(y)
	else:
		tmp = -t - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.8e+124) || !(x <= 3.3e+105))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5.8e+124) || ~((x <= 3.3e+105)))
		tmp = x * log(y);
	else
		tmp = -t - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.8e+124], N[Not[LessEqual[x, 3.3e+105]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.80000000000000043e124 or 3.29999999999999997e105 < x

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. fma-def 99.6%

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

      3. fma-neg 99.6%

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

      4. sub-neg 99.6%

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

      5. log1p-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. unsub-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 87.1%

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

   if -5.80000000000000043e124 < x < 3.29999999999999997e105

   1. Initial program 76.4%

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

      1. associate--l+ 76.4%

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

      2. fma-def 76.4%

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

      3. fma-neg 76.4%

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

      4. sub-neg 76.4%

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

      5. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. unsub-neg 99.5%

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

      4. mul-1-neg 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 79.3%

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

      1. associate-*r* 79.3%

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

      2. neg-mul-1 79.3%

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

   9. Simplified 79.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+124} \lor \neg \left(x \leq 3.3 \cdot 10^{+105}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 11: 47.6% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-8}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-78}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6e-8) (- t) (if (<= t 3e-78) (* y (- z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6e-8) {
		tmp = -t;
	} else if (t <= 3e-78) {
		tmp = y * -z;
	} else {
		tmp = -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 (t <= (-6d-8)) then
        tmp = -t
    else if (t <= 3d-78) then
        tmp = y * -z
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6e-8) {
		tmp = -t;
	} else if (t <= 3e-78) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6e-8:
		tmp = -t
	elif t <= 3e-78:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6e-8)
		tmp = Float64(-t);
	elseif (t <= 3e-78)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -6e-8)
		tmp = -t;
	elseif (t <= 3e-78)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6e-8], (-t), If[LessEqual[t, 3e-78], N[(y * (-z)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -5.99999999999999946e-8 or 2.99999999999999988e-78 < t

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. fma-def 94.4%

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

      3. sub-neg 94.4%

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

      4. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around inf 67.5%

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

      1. mul-1-neg 67.5%

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

   6. Simplified 67.5%

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

   if -5.99999999999999946e-8 < t < 2.99999999999999988e-78

   1. Initial program 70.2%

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

      1. associate--l+ 70.2%

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

      2. fma-def 70.2%

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

      3. fma-neg 70.2%

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

      4. sub-neg 70.2%

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

      5. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. unsub-neg 99.8%

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

      4. mul-1-neg 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around inf 31.8%

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

      1. associate-*r* 31.8%

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

      2. neg-mul-1 31.8%

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

   9. Simplified 31.8%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-8}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-78}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 12: 56.5% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

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 - (y * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate--l+ 83.8%

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

   2. fma-def 83.8%

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

   3. fma-neg 83.8%

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

   4. sub-neg 83.8%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 99.6%

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

   1. mul-1-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. unsub-neg 99.6%

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

   4. mul-1-neg 99.6%

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

   5. distribute-rgt-neg-in 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 58.2%

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

   1. associate-*r* 58.2%

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

   2. neg-mul-1 58.2%

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

9. Simplified 58.2%

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

10. Final simplification 58.2%

    \[\leadsto \left(-t\right) - y \cdot z \]

Alternative 13: 41.6% 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 83.8%

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

   1. +-commutative 83.8%

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

   2. fma-def 83.8%

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

   3. sub-neg 83.8%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 42.6%

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

   1. mul-1-neg 42.6%

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

6. Simplified 42.6%

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

7. Final simplification 42.6%

   \[\leadsto -t \]

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 2023275 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- 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))