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

Percentage Accurate: 85.7% → 99.8%

Time: 17.4s

Alternatives: 9

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: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

   1. associate--l+ 85.9%

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

   2. fma-define 85.9%

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

   3. sub-neg 85.9%

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

   4. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in y around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. mul-1-neg 99.7%

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

   3. unsub-neg 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 90.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-81} \lor \neg \left(t \leq 10^{-25}\right):\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -2.4e-81) (not (<= t 1e-25))) (- 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 <= -2.4e-81) || !(t <= 1e-25)) {
		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 <= (-2.4d-81)) .or. (.not. (t <= 1d-25))) 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 <= -2.4e-81) || !(t <= 1e-25)) {
		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 <= -2.4e-81) or not (t <= 1e-25):
		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 <= -2.4e-81) || !(t <= 1e-25))
		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 <= -2.4e-81) || ~((t <= 1e-25)))
		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, -2.4e-81], N[Not[LessEqual[t, 1e-25]], $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 < -2.3999999999999999e-81 or 1.00000000000000004e-25 < 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 y around 0 92.9%

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

   if -2.3999999999999999e-81 < t < 1.00000000000000004e-25

   1. Initial program 69.4%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 t around 0 97.4%

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

      1. *-commutative 97.4%

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

   8. Simplified 97.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-81} \lor \neg \left(t \leq 10^{-25}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-47} \lor \neg \left(x \leq 3.1 \cdot 10^{+37}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.1e-47) (not (<= x 3.1e+37)))
   (- (* x (log y)) t)
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e-47) || !(x <= 3.1e+37)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-3.1d-47)) .or. (.not. (x <= 3.1d+37))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y * -z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e-47) || !(x <= 3.1e+37)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.1e-47) or not (x <= 3.1e+37):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.1e-47) || !(x <= 3.1e+37))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.1e-47) || ~((x <= 3.1e+37)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.1e-47], N[Not[LessEqual[x, 3.1e+37]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0999999999999998e-47 or 3.1000000000000002e37 < x

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0 93.1%

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

   if -3.0999999999999998e-47 < x < 3.1000000000000002e37

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0 99.6%

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

      1. log1p-expm1-u 92.8%

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

      2. expm1-undefine 84.6%

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

      3. *-commutative 84.6%

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

      4. exp-to-pow 84.6%

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

   7. Applied egg-rr 84.6%

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

   8. Taylor expanded in y around inf 90.9%

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

      1. associate-*r* 90.9%

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

      2. neg-mul-1 90.9%

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

      3. *-commutative 90.9%

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

   10. Simplified 90.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-47} \lor \neg \left(x \leq 3.1 \cdot 10^{+37}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+36} \lor \neg \left(x \leq 2.2 \cdot 10^{+66}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.5e+36) (not (<= x 2.2e+66)))
   (* x (log y))
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e+36) || !(x <= 2.2e+66)) {
		tmp = x * log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-4.5d+36)) .or. (.not. (x <= 2.2d+66))) then
        tmp = x * log(y)
    else
        tmp = (y * -z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e+36) || !(x <= 2.2e+66)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.5e+36) or not (x <= 2.2e+66):
		tmp = x * math.log(y)
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.5e+36) || !(x <= 2.2e+66))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.5e+36) || ~((x <= 2.2e+66)))
		tmp = x * log(y);
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.5e+36], N[Not[LessEqual[x, 2.2e+66]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.49999999999999997e36 or 2.1999999999999998e66 < x

   1. Initial program 95.0%

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

   3. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf 83.5%

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

   if -4.49999999999999997e36 < x < 2.1999999999999998e66

   1. Initial program 79.6%

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

      1. log1p-expm1-u 88.0%

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

      2. expm1-undefine 77.6%

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

      3. *-commutative 77.6%

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

      4. exp-to-pow 77.6%

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

   7. Applied egg-rr 77.6%

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

   8. Taylor expanded in y around inf 86.5%

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

      1. associate-*r* 86.5%

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

      2. neg-mul-1 86.5%

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

      3. *-commutative 86.5%

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

   10. Simplified 86.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+36} \lor \neg \left(x \leq 2.2 \cdot 10^{+66}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% 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 85.9%

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

3. Taylor expanded in y around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. unsub-neg 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 7: 47.8% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-81} \lor \neg \left(t \leq 2.15 \cdot 10^{-11}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.8e-81) (not (<= t 2.15e-11))) (- t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.8e-81) || !(t <= 2.15e-11)) {
		tmp = -t;
	} else {
		tmp = 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 <= (-4.8d-81)) .or. (.not. (t <= 2.15d-11))) then
        tmp = -t
    else
        tmp = 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 <= -4.8e-81) || !(t <= 2.15e-11)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.8e-81) or not (t <= 2.15e-11):
		tmp = -t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.8e-81) || !(t <= 2.15e-11))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.8e-81) || ~((t <= 2.15e-11)))
		tmp = -t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.8e-81], N[Not[LessEqual[t, 2.15e-11]], $MachinePrecision]], (-t), N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.7999999999999998e-81 or 2.15000000000000001e-11 < t

   1. Initial program 93.7%

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

   3. Taylor expanded in t around inf 63.3%

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

      1. neg-mul-1 63.3%

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

   5. Simplified 63.3%

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

   if -4.7999999999999998e-81 < t < 2.15000000000000001e-11

   1. Initial program 70.4%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

   5. Simplified 99.7%

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

      1. flip-- 61.7%

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

      2. div-inv 61.5%

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

      3. difference-of-squares 61.5%

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

      4. add-sqr-sqrt 29.2%

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

      5. sqrt-unprod 61.5%

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

      6. sqr-neg 61.5%

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

      7. sqrt-unprod 32.3%

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

      8. add-sqr-sqrt 59.2%

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

      9. sub-neg 59.2%

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

      10. pow2 59.2%

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

      11. associate--l- 59.2%

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

      12. fma-define 59.2%

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

      13. add-sqr-sqrt 27.0%

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

      14. sqrt-unprod 59.2%

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

      15. sqr-neg 59.2%

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

   7. Applied egg-rr 61.5%

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

   8. Taylor expanded in y around inf 31.0%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-81} \lor \neg \left(t \leq 2.15 \cdot 10^{-11}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in y around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. unsub-neg 99.5%

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

5. Simplified 99.5%

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

   1. log1p-expm1-u 55.0%

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

   2. expm1-undefine 48.8%

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

   3. *-commutative 48.8%

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

   4. exp-to-pow 48.8%

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

7. Applied egg-rr 48.8%

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

8. Taylor expanded in y around inf 57.6%

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

   1. associate-*r* 57.6%

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

   2. neg-mul-1 57.6%

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

   3. *-commutative 57.6%

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

10. Simplified 57.6%

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

11. Final simplification 57.6%

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

Alternative 9: 42.6% 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 85.9%

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

3. Taylor expanded in t around inf 43.8%

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

   1. neg-mul-1 43.8%

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

5. Simplified 43.8%

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

6. Final simplification 43.8%

   \[\leadsto -t \]
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 2024046 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

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

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