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

Percentage Accurate: 85.5% → 99.8%

Time: 16.2s

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: 85.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

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 89.2%

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

   1. +-commutative 89.2%

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

   2. fma-def 89.2%

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

   3. sub-neg 89.2%

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

   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 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

2. Taylor expanded in y around 0 99.1%

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

   1. fma-def 99.1%

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

   2. unpow2 99.1%

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

   3. associate-*r* 99.1%

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

   4. mul-1-neg 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

   \[\leadsto \left(\mathsf{fma}\left(-0.5, z \cdot \left(y \cdot y\right), z \cdot \left(-y\right)\right) + x \cdot \log y\right) - t \]

Alternative 3: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-237} \lor \neg \left(x \leq 1.24 \cdot 10^{-141}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.6e-237) (not (<= x 1.24e-141)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.6e-237) || !(x <= 1.24e-141)) {
		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 <= -1.6e-237) || !(x <= 1.24e-141)) {
		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 <= -1.6e-237) or not (x <= 1.24e-141):
		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 <= -1.6e-237) || !(x <= 1.24e-141))
		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, -1.6e-237], N[Not[LessEqual[x, 1.24e-141]], $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 < -1.6e-237 or 1.24e-141 < x

   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 99.7%

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

   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 92.5%

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

   if -1.6e-237 < x < 1.24e-141

   1. Initial program 68.2%

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

   2. Taylor expanded in x around 0 68.1%

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

      1. sub-neg 68.1%

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

      2. mul-1-neg 68.1%

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

      3. log1p-def 100.0%

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

      4. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-237} \lor \neg \left(x \leq 1.24 \cdot 10^{-141}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 4: 88.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-237} \lor \neg \left(x \leq 1.02 \cdot 10^{-140}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.65e-237) (not (<= x 1.02e-140)))
   (- (* x (log y)) t)
   (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.65e-237) || !(x <= 1.02e-140)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.65d-237)) .or. (.not. (x <= 1.02d-140))) then
        tmp = (x * log(y)) - t
    else
        tmp = (z * -y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.65e-237) || !(x <= 1.02e-140)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.65e-237) or not (x <= 1.02e-140):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.65e-237) || !(x <= 1.02e-140))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(-y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.65e-237) || ~((x <= 1.02e-140)))
		tmp = (x * log(y)) - t;
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.65e-237], N[Not[LessEqual[x, 1.02e-140]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6500000000000001e-237 or 1.01999999999999995e-140 < x

   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 99.7%

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

   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 92.5%

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

   if -1.6500000000000001e-237 < x < 1.01999999999999995e-140

   1. Initial program 68.2%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. log-pow 99.5%

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

      4. mul-1-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. log-pow 99.5%

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

      7. *-commutative 99.5%

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

   4. Simplified 99.5%

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

      1. add-cube-cbrt 99.5%

         \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\log y \cdot x} \cdot \sqrt[3]{\log y \cdot x}\right) \cdot \sqrt[3]{\log y \cdot x}} - y \cdot z\right) - t \]

      2. pow3 99.5%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot x}\right)}^{3}} - y \cdot z\right) - t \]

      3. *-commutative 99.5%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{x \cdot \log y}}\right)}^{3} - y \cdot z\right) - t \]

   6. Applied egg-rr 99.5%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x \cdot \log y}\right)}^{3}} - y \cdot z\right) - t \]

   7. Taylor expanded in x around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-neg-out 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-237} \lor \neg \left(x \leq 1.02 \cdot 10^{-140}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]

Alternative 5: 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 89.2%

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

2. Taylor expanded in y around 0 98.6%

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

   1. +-commutative 98.6%

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

   2. *-commutative 98.6%

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

   3. log-pow 44.0%

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

   4. mul-1-neg 44.0%

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

   5. unsub-neg 44.0%

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

   6. log-pow 98.6%

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

   7. *-commutative 98.6%

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

4. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 6: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+60} \lor \neg \left(x \leq 6.4 \cdot 10^{+84}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.85e+60) (not (<= x 6.4e+84))) (* x (log y)) (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.85e+60) || !(x <= 6.4e+84)) {
		tmp = x * log(y);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.85e+60) || !(x <= 6.4e+84))
		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, -2.85e+60], N[Not[LessEqual[x, 6.4e+84]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -2.84999999999999989e60 or 6.4000000000000002e84 < x

   1. Initial program 97.1%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. fma-def 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*r* 99.1%

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

      4. mul-1-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around inf 77.8%

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

   if -2.84999999999999989e60 < x < 6.4000000000000002e84

   1. Initial program 84.1%

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

   2. Taylor expanded in y around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. *-commutative 98.4%

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

      3. log-pow 68.7%

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

      4. mul-1-neg 68.7%

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

      5. unsub-neg 68.7%

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

      6. log-pow 98.4%

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

      7. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in x around 0 77.3%

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

      1. fma-def 77.3%

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

      2. neg-mul-1 77.3%

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

   7. Simplified 77.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+60} \lor \neg \left(x \leq 6.4 \cdot 10^{+84}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 7: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+60} \lor \neg \left(x \leq 3.1 \cdot 10^{+87}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.6e+60) (not (<= x 3.1e+87)))
   (* x (log y))
   (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.6e+60) || !(x <= 3.1e+87)) {
		tmp = x * log(y);
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.6d+60)) .or. (.not. (x <= 3.1d+87))) then
        tmp = x * log(y)
    else
        tmp = (z * -y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.6e+60) || !(x <= 3.1e+87)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.6e+60) or not (x <= 3.1e+87):
		tmp = x * math.log(y)
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.6e+60) || !(x <= 3.1e+87))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(z * Float64(-y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.6e+60) || ~((x <= 3.1e+87)))
		tmp = x * log(y);
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.6e+60], N[Not[LessEqual[x, 3.1e+87]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999995e60 or 3.1e87 < x

   1. Initial program 97.1%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. fma-def 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*r* 99.1%

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

      4. mul-1-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around inf 77.8%

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

   if -1.59999999999999995e60 < x < 3.1e87

   1. Initial program 84.1%

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

   2. Taylor expanded in y around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. *-commutative 98.4%

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

      3. log-pow 68.7%

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

      4. mul-1-neg 68.7%

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

      5. unsub-neg 68.7%

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

      6. log-pow 98.4%

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

      7. *-commutative 98.4%

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

   4. Simplified 98.4%

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

      1. add-cube-cbrt 98.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\log y \cdot x} \cdot \sqrt[3]{\log y \cdot x}\right) \cdot \sqrt[3]{\log y \cdot x}} - y \cdot z\right) - t \]

      2. pow3 98.0%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot x}\right)}^{3}} - y \cdot z\right) - t \]

      3. *-commutative 98.0%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{x \cdot \log y}}\right)}^{3} - y \cdot z\right) - t \]

   6. Applied egg-rr 98.0%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x \cdot \log y}\right)}^{3}} - y \cdot z\right) - t \]

   7. Taylor expanded in x around 0 77.3%

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

      1. mul-1-neg 77.3%

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

      2. distribute-rgt-neg-out 77.3%

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

   9. Simplified 77.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+60} \lor \neg \left(x \leq 3.1 \cdot 10^{+87}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]

Alternative 8: 47.3% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -31:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-71}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -31.0) (- t) (if (<= t 5.8e-71) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -31.0) {
		tmp = -t;
	} else if (t <= 5.8e-71) {
		tmp = z * -y;
	} 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 <= (-31.0d0)) then
        tmp = -t
    else if (t <= 5.8d-71) then
        tmp = z * -y
    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 <= -31.0) {
		tmp = -t;
	} else if (t <= 5.8e-71) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -31.0:
		tmp = -t
	elif t <= 5.8e-71:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -31.0)
		tmp = Float64(-t);
	elseif (t <= 5.8e-71)
		tmp = Float64(z * Float64(-y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -31.0)
		tmp = -t;
	elseif (t <= 5.8e-71)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -31.0], (-t), If[LessEqual[t, 5.8e-71], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -31 or 5.7999999999999997e-71 < t

   1. Initial program 95.9%

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

      1. +-commutative 95.9%

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

      2. fma-def 95.9%

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

      3. sub-neg 95.9%

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

      4. log1p-def 99.8%

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

   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 71.9%

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

      1. mul-1-neg 71.9%

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

   6. Simplified 71.9%

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

   if -31 < t < 5.7999999999999997e-71

   1. Initial program 80.6%

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

   2. Taylor expanded in y around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. *-commutative 98.8%

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

      3. log-pow 31.9%

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

      4. mul-1-neg 31.9%

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

      5. unsub-neg 31.9%

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

      6. log-pow 98.8%

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

      7. *-commutative 98.8%

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

   4. Simplified 98.8%

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

   5. Taylor expanded in y around inf 22.2%

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

      1. associate-*r* 22.2%

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

      2. mul-1-neg 22.2%

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

   7. Simplified 22.2%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -31:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-71}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 9: 56.5% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

2. Taylor expanded in y around 0 98.6%

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

   1. +-commutative 98.6%

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

   2. *-commutative 98.6%

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

   3. log-pow 44.0%

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

   4. mul-1-neg 44.0%

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

   5. unsub-neg 44.0%

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

   6. log-pow 98.6%

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

   7. *-commutative 98.6%

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

4. Simplified 98.6%

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

   1. add-cube-cbrt 97.9%

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\log y \cdot x} \cdot \sqrt[3]{\log y \cdot x}\right) \cdot \sqrt[3]{\log y \cdot x}} - y \cdot z\right) - t \]

   2. pow3 97.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot x}\right)}^{3}} - y \cdot z\right) - t \]

   3. *-commutative 97.9%

      \[\leadsto \left({\left(\sqrt[3]{\color{blue}{x \cdot \log y}}\right)}^{3} - y \cdot z\right) - t \]

6. Applied egg-rr 97.9%

   \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x \cdot \log y}\right)}^{3}} - y \cdot z\right) - t \]

7. Taylor expanded in x around 0 55.4%

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

   1. mul-1-neg 55.4%

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

   2. distribute-rgt-neg-out 55.4%

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

9. Simplified 55.4%

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

10. Final simplification 55.4%

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

Alternative 10: 42.4% 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 89.2%

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

   1. +-commutative 89.2%

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

   2. fma-def 89.2%

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

   3. sub-neg 89.2%

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

   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 44.5%

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

   1. mul-1-neg 44.5%

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

6. Simplified 44.5%

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

7. Final simplification 44.5%

   \[\leadsto -t \]

Alternative 11: 2.3% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. *-commutative 89.2%

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

   2. add-sqr-sqrt 46.0%

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

   3. associate-*r* 46.0%

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

   4. fma-def 46.0%

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

   5. sub-neg 46.0%

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

   6. log1p-udef 52.5%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 45.4%

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

   9. sqr-neg 45.4%

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

   10. sqrt-unprod 45.4%

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

   11. add-sqr-sqrt 45.4%

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

3. Applied egg-rr 45.4%

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

4. Taylor expanded in y around 0 45.4%

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

   1. fma-udef 45.4%

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

   2. associate--l+ 45.4%

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

   3. associate-*r* 45.4%

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

   4. add-sqr-sqrt 87.0%

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

   5. *-commutative 87.0%

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

   6. add-cube-cbrt 86.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \log y + \left(y \cdot z - t\right) \]

   7. associate-*l* 86.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \log y\right)} + \left(y \cdot z - t\right) \]

   8. fma-def 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \log y, y \cdot z - t\right)} \]

   9. pow2 86.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt[3]{x} \cdot \log y, y \cdot z - t\right) \]

   10. fma-neg 86.2%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \log y, \color{blue}{\mathsf{fma}\left(y, z, -t\right)}\right) \]

   11. add-sqr-sqrt 42.4%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \log y, \mathsf{fma}\left(y, z, \color{blue}{\sqrt{-t} \cdot \sqrt{-t}}\right)\right) \]

   12. sqrt-unprod 49.4%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \log y, \mathsf{fma}\left(y, z, \color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}}\right)\right) \]

   13. sqr-neg 49.4%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \log y, \mathsf{fma}\left(y, z, \sqrt{\color{blue}{t \cdot t}}\right)\right) \]

   14. sqrt-unprod 19.3%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \log y, \mathsf{fma}\left(y, z, \color{blue}{\sqrt{t} \cdot \sqrt{t}}\right)\right) \]

   15. add-sqr-sqrt 42.3%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \log y, \mathsf{fma}\left(y, z, \color{blue}{t}\right)\right) \]

6. Applied egg-rr 42.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \log y, \mathsf{fma}\left(y, z, t\right)\right)} \]

7. Taylor expanded in t around inf 2.0%

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

8. Final simplification 2.0%

   \[\leadsto t \]

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 2023279 
(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))