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

Percentage Accurate: 85.0% → 99.8%

Time: 17.3s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(0 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log1p (- 0.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log1p((0.0 - y)))) - t;
}

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * Math.log1p((0.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log1p((0.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * log1p(Float64(0.0 - y)))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[1 + N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-log1p.f64 N/A

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

   3. neg-sub0 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(\color{blue}{0 - y}\right)\right) - t \]

   4. --lowering--.f64 99.8

      \[\leadsto \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(\color{blue}{0 - y}\right)\right) - t \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* x (log y))
   (* 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 ((x * log(y)) + (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(Float64(x * log(y)) + 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[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) - t \]

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right)\right) - t \]

   5. sub-neg N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)\right) - t \]

   6. metadata-eval N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right)\right) - t \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{4} \cdot y - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right)\right) - t \]

   8. sub-neg N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right)\right) - t \]

   9. *-commutative N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right)\right) - t \]

   10. metadata-eval N/A

       \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right)\right) - t \]

   11. accelerator-lowering-fma.f64 99.7

       \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right)\right) - t \]

5. Simplified 99.7%

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

Alternative 3: 99.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+ (* x (log y)) (* 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 ((x * log(y)) + (z * (y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0)))) - t;
}

Julia

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f64 99.6

      \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.3333333333333333, -0.5\right)}, -1\right)\right)\right) - t \]

5. Simplified 99.6%

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

Alternative 4: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(y, -0.5, -1\right), \mathsf{fma}\left(x, \log y, 0 - t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (* y z) (fma y -0.5 -1.0) (fma x (log y) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	return fma((y * z), fma(y, -0.5, -1.0), fma(x, log(y), (0.0 - t)));
}

Julia

function code(x, y, z, t)
	return fma(Float64(y * z), fma(y, -0.5, -1.0), fma(x, log(y), Float64(0.0 - t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * z), $MachinePrecision] * N[(y * -0.5 + -1.0), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*r* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. sub-neg N/A

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

   17. remove-double-neg N/A

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

   18. mul-1-neg N/A

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

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

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

5. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \mathsf{fma}\left(y, -0.5, -1\right), \mathsf{fma}\left(x, \log y, 0 - t\right)\right)} \]

6. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(y, -0.5, -1\right), \mathsf{fma}\left(x, \log y, 0 - t\right)\right) \]
7. Add Preprocessing

Alternative 5: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-43}:\\ \;\;\;\;0 - \mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.5e-66)
   (- (* x (log y)) t)
   (if (<= x 7.1e-43) (- 0.0 (fma z y t)) (fma x (log y) (- 0.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.5e-66) {
		tmp = (x * log(y)) - t;
	} else if (x <= 7.1e-43) {
		tmp = 0.0 - fma(z, y, t);
	} else {
		tmp = fma(x, log(y), (0.0 - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.5e-66)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (x <= 7.1e-43)
		tmp = Float64(0.0 - fma(z, y, t));
	else
		tmp = fma(x, log(y), Float64(0.0 - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.5e-66], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 7.1e-43], N[(0.0 - N[(z * y + t), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4999999999999998e-66

   1. Initial program 97.2%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f64 N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(\color{blue}{0 - y}\right)\right) - t \]

      4. --lowering--.f64 99.7

         \[\leadsto \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(\color{blue}{0 - y}\right)\right) - t \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 97.2

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

   7. Simplified 97.2%

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

   if -4.4999999999999998e-66 < x < 7.10000000000000025e-43

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
   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. --lowering--.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. Simplified 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0\right) - \mathsf{fma}\left(z, y, t\right)} \]

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(t + y \cdot z\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \left(t + y \cdot z\right)} \]

      4. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot z + t\right)} \]

      5. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{z \cdot y} + t\right) \]

      6. accelerator-lowering-fma.f64 89.1

         \[\leadsto 0 - \color{blue}{\mathsf{fma}\left(z, y, t\right)} \]

   8. Simplified 89.1%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(z, y, t\right)} \]

   if 7.10000000000000025e-43 < x

   1. Initial program 93.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-rec N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. log-lowering-log.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - t}\right) \]

      17. --lowering--.f64 91.4

          \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - t}\right) \]

   5. Simplified 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0 - t\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-41}:\\ \;\;\;\;0 - \mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -2.7e-68) t_1 (if (<= x 4.8e-41) (- 0.0 (fma z 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 <= -2.7e-68) {
		tmp = t_1;
	} else if (x <= 4.8e-41) {
		tmp = 0.0 - fma(z, 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 <= -2.7e-68)
		tmp = t_1;
	elseif (x <= 4.8e-41)
		tmp = Float64(0.0 - fma(z, 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, -2.7e-68], t$95$1, If[LessEqual[x, 4.8e-41], N[(0.0 - N[(z * y + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002e-68 or 4.80000000000000044e-41 < x

   1. Initial program 95.6%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f64 N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(\color{blue}{0 - y}\right)\right) - t \]

      4. --lowering--.f64 99.7

         \[\leadsto \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(\color{blue}{0 - y}\right)\right) - t \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 94.5

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

   7. Simplified 94.5%

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

   if -2.7000000000000002e-68 < x < 4.80000000000000044e-41

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
   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. --lowering--.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. Simplified 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0\right) - \mathsf{fma}\left(z, y, t\right)} \]

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(t + y \cdot z\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \left(t + y \cdot z\right)} \]

      4. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot z + t\right)} \]

      5. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{z \cdot y} + t\right) \]

      6. accelerator-lowering-fma.f64 89.1

         \[\leadsto 0 - \color{blue}{\mathsf{fma}\left(z, y, t\right)} \]

   8. Simplified 89.1%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(z, y, t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+78}:\\ \;\;\;\;0 - \mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.32e+28) t_1 (if (<= x 3.8e+78) (- 0.0 (fma z y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.32e+28) {
		tmp = t_1;
	} else if (x <= 3.8e+78) {
		tmp = 0.0 - fma(z, y, 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 <= -1.32e+28)
		tmp = t_1;
	elseif (x <= 3.8e+78)
		tmp = Float64(0.0 - fma(z, y, 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, -1.32e+28], t$95$1, If[LessEqual[x, 3.8e+78], N[(0.0 - N[(z * y + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3199999999999999e28 or 3.7999999999999999e78 < x

   1. Initial program 98.6%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f64 N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(\color{blue}{0 - y}\right)\right) - t \]

      4. --lowering--.f64 99.6

         \[\leadsto \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(\color{blue}{0 - y}\right)\right) - t \]

   4. Applied egg-rr 99.6%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 82.5

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

   7. Simplified 82.5%

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

   if -1.3199999999999999e28 < x < 3.7999999999999999e78

   1. Initial program 75.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
   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. --lowering--.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. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0\right) - \mathsf{fma}\left(z, y, t\right)} \]

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(t + y \cdot z\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \left(t + y \cdot z\right)} \]

      4. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot z + t\right)} \]

      5. *-commutative N/A

         \[\leadsto 0 - \left(\color{blue}{z \cdot y} + t\right) \]

      6. accelerator-lowering-fma.f64 81.8

         \[\leadsto 0 - \color{blue}{\mathsf{fma}\left(z, y, t\right)} \]

   8. Simplified 81.8%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(z, y, t\right)} \]
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 84.5%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
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. --lowering--.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. Simplified 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0\right) - \mathsf{fma}\left(z, y, t\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. log-lowering-log.f64 99.0

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 9: 48.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-41}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-29}:\\ \;\;\;\;y \cdot \left(0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.3e-41) (- 0.0 t) (if (<= t 9.8e-29) (* y (- 0.0 z)) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.3e-41) {
		tmp = 0.0 - t;
	} else if (t <= 9.8e-29) {
		tmp = y * (0.0 - z);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4.3d-41)) then
        tmp = 0.0d0 - t
    else if (t <= 9.8d-29) then
        tmp = y * (0.0d0 - z)
    else
        tmp = 0.0d0 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.3e-41) {
		tmp = 0.0 - t;
	} else if (t <= 9.8e-29) {
		tmp = y * (0.0 - z);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.3e-41:
		tmp = 0.0 - t
	elif t <= 9.8e-29:
		tmp = y * (0.0 - z)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.3e-41)
		tmp = Float64(0.0 - t);
	elseif (t <= 9.8e-29)
		tmp = Float64(y * Float64(0.0 - z));
	else
		tmp = Float64(0.0 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4.3e-41)
		tmp = 0.0 - t;
	elseif (t <= 9.8e-29)
		tmp = y * (0.0 - z);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.3e-41], N[(0.0 - t), $MachinePrecision], If[LessEqual[t, 9.8e-29], N[(y * N[(0.0 - z), $MachinePrecision]), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2999999999999999e-41 or 9.7999999999999997e-29 < t

   1. Initial program 93.6%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - t} \]

      3. --lowering--.f64 62.3

         \[\leadsto \color{blue}{0 - t} \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{0 - t} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 62.3

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

   7. Applied egg-rr 62.3%

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

   if -4.2999999999999999e-41 < t < 9.7999999999999997e-29

   1. Initial program 71.7%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. --lowering--.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. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0\right) - \mathsf{fma}\left(z, y, t\right)} \]

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - z\right)} \]

      7. --lowering--.f64 29.3

         \[\leadsto y \cdot \color{blue}{\left(0 - z\right)} \]

   8. Simplified 29.3%

      \[\leadsto \color{blue}{y \cdot \left(0 - z\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 29.3

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

   10. Applied egg-rr 29.3%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-41}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-29}:\\ \;\;\;\;y \cdot \left(0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.2% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ 0 - \mathsf{fma}\left(z, y, t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 (fma z y t)))

C

double code(double x, double y, double z, double t) {
	return 0.0 - fma(z, y, t);
}

Julia

function code(x, y, z, t)
	return Float64(0.0 - fma(z, y, t))
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - N[(z * y + t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
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. --lowering--.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. Simplified 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0\right) - \mathsf{fma}\left(z, y, t\right)} \]

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \left(t + y \cdot z\right)} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \left(t + y \cdot z\right)} \]

   4. +-commutative N/A

      \[\leadsto 0 - \color{blue}{\left(y \cdot z + t\right)} \]

   5. *-commutative N/A

      \[\leadsto 0 - \left(\color{blue}{z \cdot y} + t\right) \]

   6. accelerator-lowering-fma.f64 56.7

      \[\leadsto 0 - \color{blue}{\mathsf{fma}\left(z, y, t\right)} \]

8. Simplified 56.7%

   \[\leadsto \color{blue}{0 - \mathsf{fma}\left(z, y, t\right)} \]
9. Add Preprocessing

Alternative 11: 42.5% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ 0 - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 t))

C

double code(double x, double y, double z, double t) {
	return 0.0 - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0 - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - t;
}

Python

def code(x, y, z, t):
	return 0.0 - t

Julia

function code(x, y, z, t)
	return Float64(0.0 - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - t), $MachinePrecision]

Derivation

1. Initial program 84.5%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - t} \]

   3. --lowering--.f64 41.4

      \[\leadsto \color{blue}{0 - t} \]

5. Simplified 41.4%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 41.4

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

7. Applied egg-rr 41.4%

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

8. Final simplification 41.4%

   \[\leadsto 0 - t \]
9. Add Preprocessing

Developer Target 1: 99.6% 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 2024198 
(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))