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

Percentage Accurate: 85.0% → 99.8%

Time: 13.2s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. +-commutative 83.4%

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

   2. associate--l+ 83.4%

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

   3. fma-def 83.4%

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

   4. sub-neg 83.4%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

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 83.4%

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

   1. flip3-- 83.3%

      \[\leadsto \left(x \cdot \log y + z \cdot \log \color{blue}{\left(\frac{{1}^{3} - {y}^{3}}{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}\right)}\right) - t \]

   2. log-div 83.4%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log \left({1}^{3} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)}\right) - t \]

   3. metadata-eval 83.4%

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\log \left(\color{blue}{1} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   4. pow3 83.4%

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

   5. sub-neg 83.4%

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

   6. distribute-rgt-neg-out 83.4%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 83.1%

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

   9. sqr-neg 83.1%

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

   10. sqrt-unprod 83.1%

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

   11. add-sqr-sqrt 83.1%

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

   12. log1p-udef 83.1%

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

   13. pow3 83.1%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   14. metadata-eval 83.1%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   15. log1p-udef 99.5%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

   16. *-un-lft-identity 99.5%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

3. Applied egg-rr 99.5%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

4. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. associate-*r* 99.6%

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

   5. associate-*r* 99.6%

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

   6. neg-mul-1 99.6%

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

   7. distribute-rgt-in 99.6%

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

   8. unsub-neg 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. flip3-- 83.3%

      \[\leadsto \left(x \cdot \log y + z \cdot \log \color{blue}{\left(\frac{{1}^{3} - {y}^{3}}{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}\right)}\right) - t \]

   2. log-div 83.4%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log \left({1}^{3} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)}\right) - t \]

   3. metadata-eval 83.4%

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\log \left(\color{blue}{1} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   4. pow3 83.4%

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

   5. sub-neg 83.4%

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

   6. distribute-rgt-neg-out 83.4%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 83.1%

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

   9. sqr-neg 83.1%

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

   10. sqrt-unprod 83.1%

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

   11. add-sqr-sqrt 83.1%

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

   12. log1p-udef 83.1%

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

   13. pow3 83.1%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   14. metadata-eval 83.1%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   15. log1p-udef 99.5%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

   16. *-un-lft-identity 99.5%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

3. Applied egg-rr 99.5%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

4. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. associate-*r* 99.6%

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

   5. associate-*r* 99.6%

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

   6. neg-mul-1 99.6%

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

   7. distribute-rgt-in 99.6%

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

   8. unsub-neg 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. fma-def 99.3%

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

   3. mul-1-neg 99.3%

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

   4. *-commutative 99.3%

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

   5. distribute-rgt-neg-in 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 4: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-98} \lor \neg \left(x \leq 1.25 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8e-98) (not (<= x 1.25e-49)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8e-98) || !(x <= 1.25e-49)) {
		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 <= -8e-98) || !(x <= 1.25e-49)) {
		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 <= -8e-98) or not (x <= 1.25e-49):
		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 <= -8e-98) || !(x <= 1.25e-49))
		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, -8e-98], N[Not[LessEqual[x, 1.25e-49]], $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 < -7.99999999999999951e-98 or 1.25e-49 < x

   1. Initial program 90.4%

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

   2. Taylor expanded in x around inf 89.9%

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

   if -7.99999999999999951e-98 < x < 1.25e-49

   1. Initial program 72.5%

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

   2. Taylor expanded in x around 0 62.9%

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

      1. sub-neg 62.9%

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

      2. mul-1-neg 62.9%

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

      3. log1p-def 90.3%

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

      4. mul-1-neg 90.3%

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

   4. Simplified 90.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-98} \lor \neg \left(x \leq 1.25 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 5: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-103} \lor \neg \left(x \leq 1.05 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.8e-103) (not (<= x 1.05e-49)))
   (- (* x (log y)) t)
   (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.8e-103) || !(x <= 1.05e-49)) {
		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 <= (-3.8d-103)) .or. (.not. (x <= 1.05d-49))) 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 <= -3.8e-103) || !(x <= 1.05e-49)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.8e-103) or not (x <= 1.05e-49):
		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 <= -3.8e-103) || !(x <= 1.05e-49))
		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 <= -3.8e-103) || ~((x <= 1.05e-49)))
		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, -3.8e-103], N[Not[LessEqual[x, 1.05e-49]], $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 < -3.8000000000000001e-103 or 1.0499999999999999e-49 < x

   1. Initial program 90.4%

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

   2. Taylor expanded in x around inf 89.9%

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

   if -3.8000000000000001e-103 < x < 1.0499999999999999e-49

   1. Initial program 72.5%

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

      1. flip3-- 72.4%

         \[\leadsto \left(x \cdot \log y + z \cdot \log \color{blue}{\left(\frac{{1}^{3} - {y}^{3}}{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}\right)}\right) - t \]

      2. log-div 72.4%

         \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log \left({1}^{3} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)}\right) - t \]

      3. metadata-eval 72.4%

         \[\leadsto \left(x \cdot \log y + z \cdot \left(\log \left(\color{blue}{1} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      4. pow3 72.4%

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

      5. sub-neg 72.4%

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

      6. distribute-rgt-neg-out 72.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 71.9%

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

      9. sqr-neg 71.9%

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

      10. sqrt-unprod 71.9%

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

      11. add-sqr-sqrt 71.9%

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

      12. log1p-udef 71.9%

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

      13. pow3 71.9%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      14. metadata-eval 71.9%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      15. log1p-udef 99.2%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

      16. *-un-lft-identity 99.2%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

   3. Applied egg-rr 99.2%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   4. Taylor expanded in x around 0 62.3%

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

      1. log1p-def 62.3%

         \[\leadsto z \cdot \left(\color{blue}{\mathsf{log1p}\left({y}^{3}\right)} - \log \left(1 + \left(y + {y}^{2}\right)\right)\right) - t \]

      2. log1p-def 89.5%

         \[\leadsto z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y + {y}^{2}\right)}\right) - t \]

      3. +-commutative 89.5%

         \[\leadsto z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{{y}^{2} + y}\right)\right) - t \]

      4. unpow2 89.5%

         \[\leadsto z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{y \cdot y} + y\right)\right) - t \]

      5. fma-udef 89.5%

         \[\leadsto z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right) - t \]

   6. Simplified 89.5%

      \[\leadsto \color{blue}{z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)} - t \]

   7. Taylor expanded in y around 0 89.1%

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

      1. neg-mul-1 89.1%

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

   9. Simplified 89.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-103} \lor \neg \left(x \leq 1.05 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]

Alternative 6: 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 83.4%

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

2. Taylor expanded in y around 0 99.6%

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

3. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. mul-1-neg 99.3%

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

   3. sub-neg 99.3%

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

   4. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 7: 58.1% 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 83.4%

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

   1. flip3-- 83.3%

      \[\leadsto \left(x \cdot \log y + z \cdot \log \color{blue}{\left(\frac{{1}^{3} - {y}^{3}}{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}\right)}\right) - t \]

   2. log-div 83.4%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log \left({1}^{3} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)}\right) - t \]

   3. metadata-eval 83.4%

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\log \left(\color{blue}{1} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   4. pow3 83.4%

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

   5. sub-neg 83.4%

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

   6. distribute-rgt-neg-out 83.4%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 83.1%

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

   9. sqr-neg 83.1%

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

   10. sqrt-unprod 83.1%

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

   11. add-sqr-sqrt 83.1%

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

   12. log1p-udef 83.1%

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

   13. pow3 83.1%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   14. metadata-eval 83.1%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   15. log1p-udef 99.5%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

   16. *-un-lft-identity 99.5%

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

3. Applied egg-rr 99.5%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

4. Taylor expanded in x around 0 44.3%

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

   1. log1p-def 44.2%

      \[\leadsto z \cdot \left(\color{blue}{\mathsf{log1p}\left({y}^{3}\right)} - \log \left(1 + \left(y + {y}^{2}\right)\right)\right) - t \]

   2. log1p-def 60.0%

      \[\leadsto z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y + {y}^{2}\right)}\right) - t \]

   3. +-commutative 60.0%

      \[\leadsto z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{{y}^{2} + y}\right)\right) - t \]

   4. unpow2 60.0%

      \[\leadsto z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{y \cdot y} + y\right)\right) - t \]

   5. fma-udef 60.0%

      \[\leadsto z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right) - t \]

6. Simplified 60.0%

   \[\leadsto \color{blue}{z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)} - t \]

7. Taylor expanded in y around 0 59.8%

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

   1. neg-mul-1 59.8%

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

9. Simplified 59.8%

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

10. Final simplification 59.8%

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

Alternative 8: 43.3% accurate, 105.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 83.4%

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

   1. +-commutative 83.4%

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

   2. associate--l+ 83.4%

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

   3. fma-def 83.4%

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

   4. sub-neg 83.4%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 43.6%

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

   1. neg-mul-1 43.6%

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

6. Simplified 43.6%

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

7. Final simplification 43.6%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (*
   (- z)
   (+
    (+ (* 0.5 (* y y)) y)
    (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y)))))
  (- t (* x (log y)))))

C

double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * Math.log(y)));
}

Python

def code(x, y, z, t):
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * math.log(y)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-z) * Float64(Float64(Float64(0.5 * Float64(y * y)) + y) + Float64(Float64(0.3333333333333333 / Float64(1.0 * Float64(1.0 * 1.0))) * Float64(y * Float64(y * y))))) - Float64(t - Float64(x * log(y))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-z) * N[(N[(N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(1.0 * N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023322 
(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))