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

Percentage Accurate: 85.4% → 99.8%

Time: 16.9s

Alternatives: 17

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: 85.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip3-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. cube-neg N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-log1p.f64 N/A

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

   12. cube-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. cube-mult N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-log1p.f64 N/A

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

   19. *-lft-identity N/A

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

   20. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)\right)}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ 1.0 (/ 1.0 (fma x (log y) (fma z (log1p (- y)) (- t))))))

C

double code(double x, double y, double z, double t) {
	return 1.0 / (1.0 / fma(x, log(y), fma(z, log1p(-y), -t)));
}

Julia

function code(x, y, z, t)
	return Float64(1.0 / Float64(1.0 / fma(x, log(y), fma(z, log1p(Float64(-y)), Float64(-t)))))
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 / N[(1.0 / N[(x * N[Log[y], $MachinePrecision] + N[(z * N[Log[1 + (-y)], $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 83.9

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)\right)}}} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \left(y \cdot z\right) \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right) - z, \mathsf{fma}\left(x, \log y, -t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  y
  (- (* (* y z) (fma y -0.3333333333333333 -0.5)) z)
  (fma x (log y) (- t))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{3} \cdot \left(y \cdot z\right)\right), x \cdot \log y - t\right)} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(z \cdot y\right) \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right) - z, \mathsf{fma}\left(x, \log y, -t\right)\right)} \]

6. Final simplification 99.1%

   \[\leadsto \mathsf{fma}\left(y, \left(y \cdot z\right) \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right) - z, \mathsf{fma}\left(x, \log y, -t\right)\right) \]
7. Add Preprocessing

Alternative 4: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip3-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. cube-neg N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-log1p.f64 N/A

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

   12. cube-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. cube-mult N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-log1p.f64 N/A

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

   19. *-lft-identity N/A

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

   20. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. sub-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-neg.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 5: 88.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -4.8e+57) t_1 (if (<= x 3.3e-43) (- (* z (log1p (- y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -4.8e+57) {
		tmp = t_1;
	} else if (x <= 3.3e-43) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -4.8e+57) {
		tmp = t_1;
	} else if (x <= 3.3e-43) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -4.8e+57:
		tmp = t_1
	elif x <= 3.3e-43:
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -4.8e+57)
		tmp = t_1;
	elseif (x <= 3.3e-43)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -4.8e+57], t$95$1, If[LessEqual[x, 3.3e-43], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000009e57 or 3.30000000000000016e-43 < x

   1. Initial program 91.2%

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

   3. Taylor expanded in x around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-log.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -4.80000000000000009e57 < x < 3.30000000000000016e-43

   1. Initial program 75.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 92.0

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

   5. Applied rewrites 92.0%

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

Alternative 6: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -4.8e+57)
     t_1
     (if (<= x 3.3e-43)
       (-
        (* z (* y (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0)))
        t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -4.8e+57) {
		tmp = t_1;
	} else if (x <= 3.3e-43) {
		tmp = (z * (y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -4.8e+57)
		tmp = t_1;
	elseif (x <= 3.3e-43)
		tmp = Float64(Float64(z * Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -4.8e+57], t$95$1, If[LessEqual[x, 3.3e-43], N[(N[(z * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000009e57 or 3.30000000000000016e-43 < x

   1. Initial program 91.2%

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

   3. Taylor expanded in x around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-log.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -4.80000000000000009e57 < x < 3.30000000000000016e-43

   1. Initial program 75.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto z \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 91.2%

      \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.82 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+26}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.82e+81)
     t_1
     (if (<= x 5.8e+26)
       (-
        (* z (* y (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0)))
        t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.82e+81) {
		tmp = t_1;
	} else if (x <= 5.8e+26) {
		tmp = (z * (y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.82e+81)
		tmp = t_1;
	elseif (x <= 5.8e+26)
		tmp = Float64(Float64(z * Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.82e+81], t$95$1, If[LessEqual[x, 5.8e+26], N[(N[(z * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.82000000000000003e81 or 5.8e26 < x

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. remove-double-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)} \]

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right) \]

      11. remove-double-neg N/A

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

      12. lower-log.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -1.82000000000000003e81 < x < 5.8e26

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto z \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 88.0%

      \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

   11. lower--.f64 N/A

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

5. Applied rewrites 98.7%

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

Alternative 9: 57.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (* y (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 62.2

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in y around 0

   \[\leadsto z \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right) - t \]
7. Step-by-step derivation

8. Applied rewrites 61.6%

   \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right) - t \]
9. Add Preprocessing

Alternative 10: 57.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), y \cdot y, -y\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (fma (fma y -0.3333333333333333 -0.5) (* y y) (- y))) t))

C

double code(double x, double y, double z, double t) {
	return (z * fma(fma(y, -0.3333333333333333, -0.5), (y * y), -y)) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * fma(fma(y, -0.3333333333333333, -0.5), Float64(y * y), Float64(-y))) - t)
end

Wolfram

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

Derivation

1. Initial program 84.1%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 62.2

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 61.5%

   \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)}\right) - t \]
9. Step-by-step derivation

10. Applied rewrites 61.5%

    \[\leadsto z \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), y \cdot \color{blue}{y}, -y\right) - t \]
11. Add Preprocessing

Alternative 11: 57.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ y \cdot \mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -z\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* y (fma y (* z (fma y -0.3333333333333333 -0.5)) (- z))) t))

C

double code(double x, double y, double z, double t) {
	return (y * fma(y, (z * fma(y, -0.3333333333333333, -0.5)), -z)) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(y * fma(y, Float64(z * fma(y, -0.3333333333333333, -0.5)), Float64(-z))) - t)
end

Wolfram

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

Derivation

1. Initial program 84.1%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 62.2

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 61.1%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 61.5%

    \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -z\right)} - t \]
12. Add Preprocessing

Alternative 12: 57.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (* y (fma y (fma y -0.3333333333333333 -0.5) -1.0))) t))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 62.2

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 61.5%

   \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)}\right) - t \]
9. Add Preprocessing

Alternative 13: 48.7% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-45}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-78}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.1e-45) (- t) (if (<= t 4.8e-78) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.1e-45) {
		tmp = -t;
	} else if (t <= 4.8e-78) {
		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 <= (-1.1d-45)) then
        tmp = -t
    else if (t <= 4.8d-78) 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 <= -1.1e-45) {
		tmp = -t;
	} else if (t <= 4.8e-78) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.1e-45:
		tmp = -t
	elif t <= 4.8e-78:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.1e-45)
		tmp = Float64(-t);
	elseif (t <= 4.8e-78)
		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 <= -1.1e-45)
		tmp = -t;
	elseif (t <= 4.8e-78)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.1e-45], (-t), If[LessEqual[t, 4.8e-78], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.09999999999999997e-45 or 4.79999999999999999e-78 < t

   1. Initial program 93.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

      2. lower-neg.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -1.09999999999999997e-45 < t < 4.79999999999999999e-78

   1. Initial program 69.7%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. cube-neg N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-log1p.f64 N/A

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

      12. cube-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. cube-mult N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-log1p.f64 N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-neg.f64 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 31.4%

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

Alternative 14: 57.1% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 62.2

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 61.4%

   \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, -0.5, -1\right)}\right) - t \]
9. Add Preprocessing

Alternative 15: 56.8% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip3-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. cube-neg N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-log1p.f64 N/A

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

   12. cube-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. cube-mult N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-log1p.f64 N/A

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

   19. *-lft-identity N/A

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

   20. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. sub-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-neg.f64 98.6

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 61.1%

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

Alternative 16: 42.4% accurate, 73.3× 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 84.1%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. lower-neg.f64 45.8

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

5. Applied rewrites 45.8%

   \[\leadsto \color{blue}{-t} \]
6. Add Preprocessing

Alternative 17: 2.3% accurate, 220.0× 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 84.1%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. lower-neg.f64 45.8

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

5. Applied rewrites 45.8%

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

7. Applied rewrites 14.3%

   \[\leadsto \frac{0 - t \cdot \left(t \cdot t\right)}{\color{blue}{0 + \mathsf{fma}\left(t, t, 0 \cdot t\right)}} \]

9. Applied rewrites 2.0%

   \[\leadsto \color{blue}{t} \]
10. Add Preprocessing

Developer Target 1: 99.5% accurate, 1.3× 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 2024233 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

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