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

Percentage Accurate: 85.2% → 99.8%

Time: 13.0s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 90.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{if}\;t \leq -8.4 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- t))))
   (if (<= t -8.4e-39)
     t_1
     (if (<= t 1.3e-82) (fma (- y) z (* x (log y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, -t);
	double tmp;
	if (t <= -8.4e-39) {
		tmp = t_1;
	} else if (t <= 1.3e-82) {
		tmp = fma(-y, z, (x * log(y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-t))
	tmp = 0.0
	if (t <= -8.4e-39)
		tmp = t_1;
	elseif (t <= 1.3e-82)
		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[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]}, If[LessEqual[t, -8.4e-39], t$95$1, If[LessEqual[t, 1.3e-82], N[((-y) * z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.39999999999999973e-39 or 1.3e-82 < t

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0

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

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. lower-fma.f64 N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-log.f64 N/A

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

      15. lower-neg.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -8.39999999999999973e-39 < t < 1.3e-82

   1. Initial program 65.6%

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

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

      3. *-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. lower-fma.f64 N/A

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

      10. log-rec N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-log.f64 N/A

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

      16. sub-neg N/A

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

      17. mul-1-neg N/A

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

      18. distribute-neg-out N/A

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

      19. lower-neg.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 92.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.4 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-75}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- t))))
   (if (<= x -2.8e-69) t_1 (if (<= x 1.4e-75) (- (* z (log1p (- y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, -t);
	double tmp;
	if (x <= -2.8e-69) {
		tmp = t_1;
	} else if (x <= 1.4e-75) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-t))
	tmp = 0.0
	if (x <= -2.8e-69)
		tmp = t_1;
	elseif (x <= 1.4e-75)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]}, If[LessEqual[x, -2.8e-69], t$95$1, If[LessEqual[x, 1.4e-75], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.79999999999999979e-69 or 1.39999999999999999e-75 < x

   1. Initial program 92.5%

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

   3. Taylor expanded in y around 0

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

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. lower-fma.f64 N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-log.f64 N/A

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

      15. lower-neg.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.79999999999999979e-69 < x < 1.39999999999999999e-75

   1. Initial program 70.0%

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

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-75}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y, y, -y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- t))))
   (if (<= x -2.8e-69)
     t_1
     (if (<= x 1.4e-75)
       (-
        (*
         (fma (* (fma (fma -0.25 y -0.3333333333333333) y -0.5) y) y (- y))
         z)
        t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, -t);
	double tmp;
	if (x <= -2.8e-69) {
		tmp = t_1;
	} else if (x <= 1.4e-75) {
		tmp = (fma((fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y), y, -y) * z) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-t))
	tmp = 0.0
	if (x <= -2.8e-69)
		tmp = t_1;
	elseif (x <= 1.4e-75)
		tmp = Float64(Float64(fma(Float64(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y), y, Float64(-y)) * z) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]}, If[LessEqual[x, -2.8e-69], t$95$1, If[LessEqual[x, 1.4e-75], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y), $MachinePrecision] * y + (-y)), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.79999999999999979e-69 or 1.39999999999999999e-75 < x

   1. Initial program 92.5%

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

   3. Taylor expanded in y around 0

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

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. lower-fma.f64 N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-log.f64 N/A

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

      15. lower-neg.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.79999999999999979e-69 < x < 1.39999999999999999e-75

   1. Initial program 70.0%

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

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.3%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t \]
   9. Step-by-step derivation

   10. Applied rewrites 92.3%

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

Alternative 5: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -14000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y, y, -y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -14000.0)
     t_1
     (if (<= x 1.32e+118)
       (-
        (*
         (fma (* (fma (fma -0.25 y -0.3333333333333333) y -0.5) y) y (- y))
         z)
        t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -14000.0) {
		tmp = t_1;
	} else if (x <= 1.32e+118) {
		tmp = (fma((fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y), y, -y) * z) - 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 <= -14000.0)
		tmp = t_1;
	elseif (x <= 1.32e+118)
		tmp = Float64(Float64(fma(Float64(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y), y, Float64(-y)) * z) - 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, -14000.0], t$95$1, If[LessEqual[x, 1.32e+118], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y), $MachinePrecision] * y + (-y)), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -14000 or 1.3199999999999999e118 < x

   1. Initial program 94.3%

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

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if -14000 < x < 1.3199999999999999e118

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t \]
   9. Step-by-step derivation

   10. Applied rewrites 80.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14000:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y, y, -y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

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

   3. *-commutative N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. log-rec N/A

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

   9. lower-fma.f64 N/A

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

   10. log-rec N/A

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

   11. mul-1-neg N/A

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

   12. mul-1-neg N/A

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

   13. mul-1-neg N/A

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

   14. remove-double-neg N/A

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

   15. lower-log.f64 N/A

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

   16. sub-neg N/A

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

   17. mul-1-neg N/A

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

   18. distribute-neg-out N/A

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

   19. lower-neg.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 7: 58.2% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.1

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

5. Applied rewrites 59.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 59.0%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t \]
9. Step-by-step derivation

10. Applied rewrites 59.0%

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

Alternative 8: 58.1% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.1

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

5. Applied rewrites 59.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 59.0%

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

Alternative 9: 58.1% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.1

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

5. Applied rewrites 59.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 59.0%

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

Alternative 10: 48.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-46}:\\ \;\;\;\;-z \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.6e-45) (- t) (if (<= t 3.2e-46) (- (* z y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e-45) {
		tmp = -t;
	} else if (t <= 3.2e-46) {
		tmp = -(z * y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.6d-45)) then
        tmp = -t
    else if (t <= 3.2d-46) then
        tmp = -(z * y)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e-45) {
		tmp = -t;
	} else if (t <= 3.2e-46) {
		tmp = -(z * y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.6e-45:
		tmp = -t
	elif t <= 3.2e-46:
		tmp = -(z * y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.6e-45)
		tmp = Float64(-t);
	elseif (t <= 3.2e-46)
		tmp = Float64(-Float64(z * y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.6e-45)
		tmp = -t;
	elseif (t <= 3.2e-46)
		tmp = -(z * y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.6e-45], (-t), If[LessEqual[t, 3.2e-46], (-N[(z * y), $MachinePrecision]), (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -2.59999999999999987e-45 or 3.1999999999999999e-46 < t

   1. Initial program 96.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. lower-neg.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -2.59999999999999987e-45 < t < 3.1999999999999999e-46

   1. Initial program 67.7%

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

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

      3. *-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. lower-fma.f64 N/A

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

      10. log-rec N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-log.f64 N/A

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

      16. sub-neg N/A

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

      17. mul-1-neg N/A

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

      18. distribute-neg-out N/A

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

      19. lower-neg.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto -y \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 33.0%

       \[\leadsto -z \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.0% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.1

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

5. Applied rewrites 59.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 58.9%

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

Alternative 12: 57.7% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

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

   3. *-commutative N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. log-rec N/A

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

   9. lower-fma.f64 N/A

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

   10. log-rec N/A

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

   11. mul-1-neg N/A

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

   12. mul-1-neg N/A

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

   13. mul-1-neg N/A

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

   14. remove-double-neg N/A

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

   15. lower-log.f64 N/A

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

   16. sub-neg N/A

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

   17. mul-1-neg N/A

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

   18. distribute-neg-out N/A

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

   19. lower-neg.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 58.8%

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

Alternative 13: 43.1% 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 83.7%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 43.6

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

5. Applied rewrites 43.6%

   \[\leadsto \color{blue}{-t} \]
6. 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 2024294 
(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))