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

Percentage Accurate: 85.6% → 99.6%

Time: 17.9s

Alternatives: 14

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: 85.6% 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: 99.6% accurate, 1.6× speedup

Math

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

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 88.1%

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

3. Taylor expanded in y around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

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

   3. mul-1-neg N/A

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

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

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

   5. neg-mul-1 N/A

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 88.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) (- t))))
   (if (<= t -1.7e-239)
     t_1
     (if (<= t 1.15e-129) (fma (- y) z (* x (log y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, log(y), -t);
	double tmp;
	if (t <= -1.7e-239) {
		tmp = t_1;
	} else if (t <= 1.15e-129) {
		tmp = fma(-y, z, (x * log(y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, log(y), Float64(-t))
	tmp = 0.0
	if (t <= -1.7e-239)
		tmp = t_1;
	elseif (t <= 1.15e-129)
		tmp = fma(Float64(-y), z, Float64(x * log(y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[t, -1.7e-239], t$95$1, If[LessEqual[t, 1.15e-129], N[((-y) * z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7e-239 or 1.15e-129 < t

   1. Initial program 92.3%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. lower-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. lower-log.f64 N/A

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

      16. lower-neg.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -1.7e-239 < t < 1.15e-129

   1. Initial program 75.3%

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

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {\log y}^{2}, \frac{x}{x \cdot \log y - z \cdot \mathsf{log1p}\left(-y\right)}, -\frac{{\left(z \cdot \mathsf{log1p}\left(-y\right)\right)}^{2}}{x \cdot \log y - z \cdot \mathsf{log1p}\left(-y\right)}\right)} - t \]

   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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), z, x \cdot \log y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.6%

       \[\leadsto \mathsf{fma}\left(-y, z, x \cdot \log y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-203}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right), -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) (- t))))
   (if (<= x -8.2e-99)
     t_1
     (if (<= x 1.45e-203)
       (fma z (* y (fma y (fma y -0.3333333333333333 -0.5) -1.0)) (- t))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, log(y), -t);
	double tmp;
	if (x <= -8.2e-99) {
		tmp = t_1;
	} else if (x <= 1.45e-203) {
		tmp = fma(z, (y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0)), -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, log(y), Float64(-t))
	tmp = 0.0
	if (x <= -8.2e-99)
		tmp = t_1;
	elseif (x <= 1.45e-203)
		tmp = fma(z, Float64(y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0)), Float64(-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] + (-t)), $MachinePrecision]}, If[LessEqual[x, -8.2e-99], t$95$1, If[LessEqual[x, 1.45e-203], N[(z * N[(y * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.20000000000000057e-99 or 1.4499999999999999e-203 < x

   1. Initial program 92.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \log y - t} \]
   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. lower-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. lower-log.f64 N/A

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

      16. lower-neg.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if -8.20000000000000057e-99 < x < 1.4499999999999999e-203

   1. Initial program 76.4%

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

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 96.0%

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

Alternative 5: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right), -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -4.2e+79)
     t_1
     (if (<= x 1.35e-16)
       (fma z (* y (fma y (fma y -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 <= -4.2e+79) {
		tmp = t_1;
	} else if (x <= 1.35e-16) {
		tmp = fma(z, (y * fma(y, fma(y, -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 <= -4.2e+79)
		tmp = t_1;
	elseif (x <= 1.35e-16)
		tmp = fma(z, Float64(y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0)), Float64(-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, -4.2e+79], t$95$1, If[LessEqual[x, 1.35e-16], N[(z * N[(y * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000016e79 or 1.35e-16 < x

   1. Initial program 95.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

   if -4.20000000000000016e79 < x < 1.35e-16

   1. Initial program 81.7%

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

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-neg.f64 81.5

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.5%

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

Alternative 6: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

4. Applied rewrites 94.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {\log y}^{2}, \frac{x}{x \cdot \log y - z \cdot \mathsf{log1p}\left(-y\right)}, -\frac{{\left(z \cdot \mathsf{log1p}\left(-y\right)\right)}^{2}}{x \cdot \log y - z \cdot \mathsf{log1p}\left(-y\right)}\right)} - t \]

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. associate-*r* N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-neg.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 7: 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 88.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 99.6%

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

Alternative 8: 57.3% accurate, 6.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  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 fma(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 fma(z, Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0)), Float64(-t))
end

Wolfram

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

Derivation

1. Initial program 88.1%

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

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-neg.f64 56.0

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

5. Applied rewrites 56.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 56.0%

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

Alternative 9: 57.2% accurate, 8.5× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-neg.f64 56.0

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

5. Applied rewrites 56.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 56.0%

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

Alternative 10: 46.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-252}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.7e-252) (- t) (if (<= t 6.8e-130) (* y (- z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.7e-252) {
		tmp = -t;
	} else if (t <= 6.8e-130) {
		tmp = y * -z;
	} 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 <= (-1.7d-252)) then
        tmp = -t
    else if (t <= 6.8d-130) then
        tmp = y * -z
    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 <= -1.7e-252) {
		tmp = -t;
	} else if (t <= 6.8e-130) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.7e-252:
		tmp = -t
	elif t <= 6.8e-130:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.7e-252)
		tmp = Float64(-t);
	elseif (t <= 6.8e-130)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.7e-252)
		tmp = -t;
	elseif (t <= 6.8e-130)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.7e-252], (-t), If[LessEqual[t, 6.8e-130], N[(y * (-z)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7e-252 or 6.8000000000000001e-130 < t

   1. Initial program 92.4%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

   if -1.7e-252 < t < 6.8000000000000001e-130

   1. Initial program 73.2%

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

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {\log y}^{2}, \frac{x}{x \cdot \log y - z \cdot \mathsf{log1p}\left(-y\right)}, -\frac{{\left(z \cdot \mathsf{log1p}\left(-y\right)\right)}^{2}}{x \cdot \log y - z \cdot \mathsf{log1p}\left(-y\right)}\right)} - t \]

   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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 29.1%

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

Alternative 11: 57.1% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-neg.f64 56.0

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

5. Applied rewrites 56.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 55.9%

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

Alternative 12: 56.8% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

4. Applied rewrites 94.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {\log y}^{2}, \frac{x}{x \cdot \log y - z \cdot \mathsf{log1p}\left(-y\right)}, -\frac{{\left(z \cdot \mathsf{log1p}\left(-y\right)\right)}^{2}}{x \cdot \log y - z \cdot \mathsf{log1p}\left(-y\right)}\right)} - t \]

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. associate-*r* N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-neg.f64 99.6

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

7. Applied rewrites 99.6%

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

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 55.8%

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

Alternative 13: 42.7% 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 88.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 44.3

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

5. Applied rewrites 44.3%

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

Alternative 14: 2.2% 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 88.1%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 44.3

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

5. Applied rewrites 44.3%

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

7. Applied rewrites 13.1%

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

9. Applied rewrites 2.2%

   \[\leadsto \color{blue}{t} \]
10. 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 2024220 
(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))