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

Percentage Accurate: 85.2% → 99.8%

Time: 10.4s

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: 85.2% 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, 0.7× 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 87.5%

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

   1. +-commutative 87.5%

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

   2. fma-def 87.5%

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

   3. sub-neg 87.5%

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

   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 87.5%

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

2. Taylor expanded in y around 0 99.7%

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

   1. fma-def 99.7%

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

   2. unpow2 99.7%

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

   3. associate-*r* 99.7%

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

   4. mul-1-neg 99.7%

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

4. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-69} \lor \neg \left(x \leq 5.4 \cdot 10^{-37}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.4e-69) (not (<= x 5.4e-37)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e-69) || !(x <= 5.4e-37)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e-69) || !(x <= 5.4e-37)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.4e-69) or not (x <= 5.4e-37):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.4e-69) || !(x <= 5.4e-37))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.4e-69], N[Not[LessEqual[x, 5.4e-37]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e-69 or 5.40000000000000032e-37 < x

   1. Initial program 94.8%

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

   2. Taylor expanded in y around 0 99.6%

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

      1. fma-def 99.6%

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

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. mul-1-neg 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around inf 93.7%

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

   if -2.4000000000000001e-69 < x < 5.40000000000000032e-37

   1. Initial program 77.2%

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

   2. Taylor expanded in x around 0 71.9%

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

      1. sub-neg 71.9%

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

      2. mul-1-neg 71.9%

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

      3. log1p-def 94.6%

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

      4. mul-1-neg 94.6%

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

   4. Simplified 94.6%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-69} \lor \neg \left(x \leq 5.4 \cdot 10^{-37}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 4: 90.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-69} \lor \neg \left(x \leq 3.3 \cdot 10^{-37}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(y \cdot -0.5 + -1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.4e-69) (not (<= x 3.3e-37)))
   (- (* x (log y)) t)
   (- (* (* z y) (+ (* y -0.5) -1.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e-69) || !(x <= 3.3e-37)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = ((z * y) * ((y * -0.5) + -1.0)) - 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 <= (-2.4d-69)) .or. (.not. (x <= 3.3d-37))) then
        tmp = (x * log(y)) - t
    else
        tmp = ((z * y) * ((y * (-0.5d0)) + (-1.0d0))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e-69) || !(x <= 3.3e-37)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = ((z * y) * ((y * -0.5) + -1.0)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.4e-69) or not (x <= 3.3e-37):
		tmp = (x * math.log(y)) - t
	else:
		tmp = ((z * y) * ((y * -0.5) + -1.0)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.4e-69) || !(x <= 3.3e-37))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(z * y) * Float64(Float64(y * -0.5) + -1.0)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.4e-69) || ~((x <= 3.3e-37)))
		tmp = (x * log(y)) - t;
	else
		tmp = ((z * y) * ((y * -0.5) + -1.0)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.4e-69], N[Not[LessEqual[x, 3.3e-37]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * N[(N[(y * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e-69 or 3.29999999999999982e-37 < x

   1. Initial program 94.8%

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

   2. Taylor expanded in y around 0 99.6%

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

      1. fma-def 99.6%

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

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. mul-1-neg 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around inf 93.7%

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

   if -2.4000000000000001e-69 < x < 3.29999999999999982e-37

   1. Initial program 77.2%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. fma-def 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

      4. mul-1-neg 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 94.5%

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

      1. unpow2 94.5%

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

      2. associate-*r* 94.5%

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

      3. mul-1-neg 94.5%

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

      4. sub-neg 94.5%

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

      5. associate-*r* 94.5%

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

      6. *-lft-identity 94.5%

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

      7. distribute-rgt-out-- 94.5%

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

      8. *-commutative 94.5%

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

   7. Simplified 94.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-69} \lor \neg \left(x \leq 3.3 \cdot 10^{-37}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(y \cdot -0.5 + -1\right) - t\\ \end{array} \]

Alternative 5: 99.2% 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 87.5%

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

2. Taylor expanded in y around 0 99.2%

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

   1. +-commutative 99.2%

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

   2. fma-def 99.2%

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

   3. mul-1-neg 99.2%

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

   4. fma-neg 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 6: 57.5% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (z * (-y - ((y * y) * 0.5d0))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

2. Taylor expanded in y around 0 99.7%

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

   1. fma-def 99.7%

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

   2. unpow2 99.7%

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

   3. associate-*r* 99.7%

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

   4. mul-1-neg 99.7%

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

4. Simplified 99.7%

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

5. Taylor expanded in z around -inf 58.7%

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

   1. mul-1-neg 58.7%

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

   2. *-commutative 58.7%

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

   3. *-commutative 58.7%

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

   4. unpow2 58.7%

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

7. Simplified 58.7%

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

8. Final simplification 58.7%

   \[\leadsto z \cdot \left(\left(-y\right) - \left(y \cdot y\right) \cdot 0.5\right) - t \]

Alternative 7: 57.5% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((z * y) * ((y * (-0.5d0)) + (-1.0d0))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

2. Taylor expanded in y around 0 99.7%

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

   1. fma-def 99.7%

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

   2. unpow2 99.7%

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

   3. associate-*r* 99.7%

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

   4. mul-1-neg 99.7%

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

4. Simplified 99.7%

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

5. Taylor expanded in x around 0 58.8%

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

   1. unpow2 58.8%

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

   2. associate-*r* 58.8%

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

   3. mul-1-neg 58.8%

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

   4. sub-neg 58.8%

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

   5. associate-*r* 58.8%

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

   6. *-lft-identity 58.8%

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

   7. distribute-rgt-out-- 58.7%

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

   8. *-commutative 58.7%

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

7. Simplified 58.7%

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

8. Final simplification 58.7%

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

Alternative 8: 57.2% 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 87.5%

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

2. Taylor expanded in x around 0 46.6%

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

   1. sub-neg 46.6%

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

   2. mul-1-neg 46.6%

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

   3. log1p-def 58.9%

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

   4. mul-1-neg 58.9%

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

4. Simplified 58.9%

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

5. Taylor expanded in y around 0 58.2%

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

   1. neg-mul-1 58.2%

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

   2. mul-1-neg 58.2%

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

   3. unsub-neg 58.2%

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

7. Simplified 58.2%

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

8. Final simplification 58.2%

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

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 87.5%

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

2. Taylor expanded in x around 0 46.6%

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

   1. sub-neg 46.6%

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

   2. mul-1-neg 46.6%

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

   3. log1p-def 58.9%

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

   4. mul-1-neg 58.9%

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

4. Simplified 58.9%

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

5. Taylor expanded in z around 0 45.4%

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

   1. neg-mul-1 45.4%

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

7. Simplified 45.4%

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

8. Final simplification 45.4%

   \[\leadsto -t \]

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 2023238 
(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))