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

Percentage Accurate: 85.2% → 99.8%

Time: 13.7s

Alternatives: 11

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: 85.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

   1. +-commutative 83.6%

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

   2. associate--l+ 83.6%

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

   3. fma-def 83.6%

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

   4. sub-neg 83.6%

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

   5. log1p-def 99.8%

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

   6. fma-neg 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

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 83.6%

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

   1. +-commutative 83.6%

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

   2. associate--l+ 83.6%

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

   3. fma-def 83.6%

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

   4. sub-neg 83.6%

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

   5. log1p-def 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. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + \left(-0.5 \cdot \left(z \cdot {y}^{2}\right) - z \cdot y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (- (* -0.5 (* z (pow y 2.0))) (* z y))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + ((-0.5 * (z * pow(y, 2.0))) - (z * 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)) + (((-0.5d0) * (z * (y ** 2.0d0))) - (z * y))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + ((-0.5 * (z * Math.pow(y, 2.0))) - (z * y))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + ((-0.5 * (z * math.pow(y, 2.0))) - (z * y))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(Float64(-0.5 * Float64(z * (y ^ 2.0))) - Float64(z * y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + ((-0.5 * (z * (y ^ 2.0))) - (z * y))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(z * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 83.6%

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

3. Taylor expanded in y around 0 99.5%

   \[\leadsto \left(x \cdot \log y + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot \left({y}^{2} \cdot z\right)\right)}\right) - t \]

4. Final simplification 99.5%

   \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \left(z \cdot {y}^{2}\right) - z \cdot y\right)\right) - t \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-54} \lor \neg \left(x \leq 5.5 \cdot 10^{-41}\right):\\ \;\;\;\;\left(-t\right) - x \cdot \log \left(\frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.2e-54) (not (<= x 5.5e-41)))
   (- (- t) (* x (log (/ 1.0 y))))
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.2e-54) || !(x <= 5.5e-41)) {
		tmp = -t - (x * log((1.0 / y)));
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.2e-54) || !(x <= 5.5e-41))
		tmp = Float64(Float64(-t) - Float64(x * log(Float64(1.0 / y))));
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.2e-54], N[Not[LessEqual[x, 5.5e-41]], $MachinePrecision]], N[((-t) - N[(x * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e-54 or 5.50000000000000022e-41 < x

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0 93.3%

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

   4. Taylor expanded in y around inf 93.3%

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

   if -4.2e-54 < x < 5.50000000000000022e-41

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 95.2%

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

      1. mul-1-neg 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-udef 95.2%

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

   8. Simplified 95.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-54} \lor \neg \left(x \leq 5.5 \cdot 10^{-41}\right):\\ \;\;\;\;\left(-t\right) - x \cdot \log \left(\frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-55} \lor \neg \left(x \leq 3.2 \cdot 10^{-43}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.95e-55) (not (<= x 3.2e-43)))
   (- (* x (log y)) t)
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.95e-55) || !(x <= 3.2e-43)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.95e-55) || !(x <= 3.2e-43))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.95e-55], N[Not[LessEqual[x, 3.2e-43]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -2.9499999999999999e-55 or 3.19999999999999985e-43 < x

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0 93.3%

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

   if -2.9499999999999999e-55 < x < 3.19999999999999985e-43

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 95.2%

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

      1. mul-1-neg 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-udef 95.2%

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

   8. Simplified 95.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-55} \lor \neg \left(x \leq 3.2 \cdot 10^{-43}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+144} \lor \neg \left(x \leq 5.4 \cdot 10^{+78}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7.2e+144) (not (<= x 5.4e+78))) (* x (log y)) (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7.2e+144) || !(x <= 5.4e+78)) {
		tmp = x * log(y);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7.2e+144) || !(x <= 5.4e+78))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7.2e+144], N[Not[LessEqual[x, 5.4e+78]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -7.1999999999999995e144 or 5.40000000000000009e78 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 99.5%

      \[\leadsto \left(x \cdot \log y + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot \left({y}^{2} \cdot z\right)\right)}\right) - t \]

   4. Taylor expanded in x around inf 85.1%

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

   if -7.1999999999999995e144 < x < 5.40000000000000009e78

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. mul-1-neg 99.2%

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

      3. unsub-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0 81.7%

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

      1. mul-1-neg 81.7%

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

      2. +-commutative 81.7%

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

      3. fma-udef 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+144} \lor \neg \left(x \leq 5.4 \cdot 10^{+78}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+144} \lor \neg \left(x \leq 1.42 \cdot 10^{+79}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.5e+144) (not (<= x 1.42e+79)))
   (* x (log y))
   (- (- t) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.5e+144) || !(x <= 1.42e+79)) {
		tmp = x * log(y);
	} else {
		tmp = -t - (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 ((x <= (-3.5d+144)) .or. (.not. (x <= 1.42d+79))) then
        tmp = x * log(y)
    else
        tmp = -t - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.5e+144) || !(x <= 1.42e+79)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.5e+144) or not (x <= 1.42e+79):
		tmp = x * math.log(y)
	else:
		tmp = -t - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.5e+144) || !(x <= 1.42e+79))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.5e+144) || ~((x <= 1.42e+79)))
		tmp = x * log(y);
	else
		tmp = -t - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.5e+144], N[Not[LessEqual[x, 1.42e+79]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4999999999999998e144 or 1.41999999999999998e79 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 99.5%

      \[\leadsto \left(x \cdot \log y + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot \left({y}^{2} \cdot z\right)\right)}\right) - t \]

   4. Taylor expanded in x around inf 85.1%

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

   if -3.4999999999999998e144 < x < 1.41999999999999998e79

   1. Initial program 76.3%

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

   3. Taylor expanded in x around 0 59.5%

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

      1. fma-neg 59.5%

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

      2. sub-neg 59.5%

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

      3. mul-1-neg 59.5%

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

      4. log1p-def 82.4%

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

      5. mul-1-neg 82.4%

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

   5. Simplified 82.4%

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

   6. Taylor expanded in y around 0 81.7%

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

      1. mul-1-neg 81.7%

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

      2. +-commutative 81.7%

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

      3. unsub-neg 81.7%

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

      4. mul-1-neg 81.7%

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

      5. distribute-rgt-neg-in 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+144} \lor \neg \left(x \leq 1.42 \cdot 10^{+79}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

3. Taylor expanded in y around 0 99.2%

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

   1. +-commutative 99.2%

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

   2. mul-1-neg 99.2%

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

   3. unsub-neg 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 9: 48.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-90} \lor \neg \left(t \leq 5 \cdot 10^{-29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.35e-90) (not (<= t 5e-29))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.35e-90) || !(t <= 5e-29)) {
		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 <= (-1.35d-90)) .or. (.not. (t <= 5d-29))) 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 <= -1.35e-90) || !(t <= 5e-29)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.35e-90) or not (t <= 5e-29):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.35e-90) || !(t <= 5e-29))
		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 <= -1.35e-90) || ~((t <= 5e-29)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.35e-90], N[Not[LessEqual[t, 5e-29]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.34999999999999998e-90 or 4.99999999999999986e-29 < t

   1. Initial program 93.8%

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

   3. Taylor expanded in t around inf 66.6%

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

      1. mul-1-neg 66.6%

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

   5. Simplified 66.6%

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

   if -1.34999999999999998e-90 < t < 4.99999999999999986e-29

   1. Initial program 66.6%

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

   3. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 34.7%

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

      1. mul-1-neg 34.7%

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

      2. *-commutative 34.7%

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

      3. distribute-rgt-neg-in 34.7%

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

   8. Simplified 34.7%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-90} \lor \neg \left(t \leq 5 \cdot 10^{-29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% 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 83.6%

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

3. Taylor expanded in x around 0 45.2%

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

   1. fma-neg 45.2%

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

   2. sub-neg 45.2%

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

   3. mul-1-neg 45.2%

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

   4. log1p-def 60.9%

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

   5. mul-1-neg 60.9%

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

5. Simplified 60.9%

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

6. Taylor expanded in y around 0 60.3%

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

   1. mul-1-neg 60.3%

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

   2. +-commutative 60.3%

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

   3. unsub-neg 60.3%

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

   4. mul-1-neg 60.3%

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

   5. distribute-rgt-neg-in 60.3%

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

8. Simplified 60.3%

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

9. Final simplification 60.3%

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

Alternative 11: 42.9% 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 83.6%

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

3. Taylor expanded in t around inf 44.3%

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

   1. mul-1-neg 44.3%

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

5. Simplified 44.3%

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

6. Final simplification 44.3%

   \[\leadsto -t \]
7. 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 2024036 
(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))