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

Percentage Accurate: 85.2% → 99.5%

Time: 14.8s

Alternatives: 10

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: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

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

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

   13. neg-mul-1 N/A

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

   14. mul-1-neg N/A

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

   15. log-rec N/A

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

   16. lower-fma.f64 N/A

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

5. Simplified 99.8%

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

Alternative 2: 87.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - z \cdot y\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) (* z y))))
   (if (<= z -5.8e+137) t_1 (if (<= z 4.8e+174) (fma x (log y) (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - (z * y);
	double tmp;
	if (z <= -5.8e+137) {
		tmp = t_1;
	} else if (z <= 4.8e+174) {
		tmp = fma(x, log(y), -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - Float64(z * y))
	tmp = 0.0
	if (z <= -5.8e+137)
		tmp = t_1;
	elseif (z <= 4.8e+174)
		tmp = fma(x, log(y), Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.8e+137], t$95$1, If[LessEqual[z, 4.8e+174], N[(x * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.79999999999999969e137 or 4.7999999999999996e174 < z

   1. Initial program 49.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.3

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

   8. Simplified 99.3%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-log.f64 N/A

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

       7. lower-*.f64 85.5

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

   11. Simplified 85.5%

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

   if -5.79999999999999969e137 < z < 4.7999999999999996e174

   1. Initial program 97.3%

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. lower-fma.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-neg.f64 97.3

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

   5. Simplified 97.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+137}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, -0.5, -1\right), -t\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.3e+168)
   (fma y (* z (fma y -0.5 -1.0)) (- t))
   (if (<= z 2.2e+169) (fma x (log y) (- t)) (- (fma y z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e+168) {
		tmp = fma(y, (z * fma(y, -0.5, -1.0)), -t);
	} else if (z <= 2.2e+169) {
		tmp = fma(x, log(y), -t);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.3e+168)
		tmp = fma(y, Float64(z * fma(y, -0.5, -1.0)), Float64(-t));
	elseif (z <= 2.2e+169)
		tmp = fma(x, log(y), Float64(-t));
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.3e+168], N[(y * N[(z * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[z, 2.2e+169], N[(x * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e168

   1. Initial program 47.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 69.0

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

   8. Simplified 69.0%

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

   if -1.3e168 < z < 2.2e169

   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 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. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. lower-fma.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-neg.f64 96.5

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

   5. Simplified 96.5%

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

   if 2.2e169 < z

   1. Initial program 54.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.9

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. distribute-neg-out N/A

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

       4. remove-double-neg N/A

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

       5. distribute-neg-out N/A

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

       6. mul-1-neg N/A

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

       7. sub-neg N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

       12. mul-1-neg N/A

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

       13. distribute-neg-out N/A

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

       14. remove-double-neg N/A

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

       15. lower-fma.f64 75.5

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

   11. Simplified 75.5%

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

Alternative 4: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+58}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.35e+132) t_1 (if (<= x 2.35e+58) (- (fma 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 <= -1.35e+132) {
		tmp = t_1;
	} else if (x <= 2.35e+58) {
		tmp = -fma(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 <= -1.35e+132)
		tmp = t_1;
	elseif (x <= 2.35e+58)
		tmp = Float64(-fma(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, -1.35e+132], t$95$1, If[LessEqual[x, 2.35e+58], (-N[(y * z + t), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e132 or 2.34999999999999986e58 < x

   1. Initial program 97.4%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 86.3

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

   5. Simplified 86.3%

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

   if -1.35e132 < x < 2.34999999999999986e58

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.9

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. distribute-neg-out N/A

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

       4. remove-double-neg N/A

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

       5. distribute-neg-out N/A

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

       6. mul-1-neg N/A

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

       7. sub-neg N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

       12. mul-1-neg N/A

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

       13. distribute-neg-out N/A

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

       14. remove-double-neg N/A

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

       15. lower-fma.f64 78.4

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

   11. Simplified 78.4%

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

Alternative 5: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

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

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

   13. neg-mul-1 N/A

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

   14. mul-1-neg N/A

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

   15. log-rec N/A

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

   16. lower-fma.f64 N/A

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

5. Simplified 99.8%

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

6. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.7

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

8. Simplified 99.7%

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

Alternative 6: 99.2% 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 88.4%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

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

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

   11. 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. Simplified 99.7%

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

Alternative 7: 46.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -7 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+174}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- y))))
   (if (<= z -7e+137) t_1 (if (<= z 4.8e+174) (- t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * -y;
	double tmp;
	if (z <= -7e+137) {
		tmp = t_1;
	} else if (z <= 4.8e+174) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = z * -y
    if (z <= (-7d+137)) then
        tmp = t_1
    else if (z <= 4.8d+174) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * -y;
	double tmp;
	if (z <= -7e+137) {
		tmp = t_1;
	} else if (z <= 4.8e+174) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * -y
	tmp = 0
	if z <= -7e+137:
		tmp = t_1
	elif z <= 4.8e+174:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(-y))
	tmp = 0.0
	if (z <= -7e+137)
		tmp = t_1;
	elseif (z <= 4.8e+174)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * -y;
	tmp = 0.0;
	if (z <= -7e+137)
		tmp = t_1;
	elseif (z <= 4.8e+174)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * (-y)), $MachinePrecision]}, If[LessEqual[z, -7e+137], t$95$1, If[LessEqual[z, 4.8e+174], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.0000000000000002e137 or 4.7999999999999996e174 < z

   1. Initial program 49.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.3

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

   8. Simplified 99.3%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 51.7

          \[\leadsto -\color{blue}{y \cdot z} \]

   11. Simplified 51.7%

       \[\leadsto \color{blue}{-y \cdot z} \]

   if -7.0000000000000002e137 < z < 4.7999999999999996e174

   1. Initial program 97.3%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 51.7

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

   5. Simplified 51.7%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+174}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.5% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

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

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

   13. neg-mul-1 N/A

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

   14. mul-1-neg N/A

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

   15. log-rec N/A

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

   16. lower-fma.f64 N/A

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 56.4

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

8. Simplified 56.4%

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

Alternative 9: 58.2% 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 88.4%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

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

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

   13. neg-mul-1 N/A

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

   14. mul-1-neg N/A

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

   15. log-rec N/A

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

   16. lower-fma.f64 N/A

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

5. Simplified 99.8%

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

6. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.7

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

8. Simplified 99.7%

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

9. Taylor expanded in x around 0

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

    1. sub-neg N/A

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

    2. mul-1-neg N/A

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

    3. distribute-neg-out N/A

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

    4. remove-double-neg N/A

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

    5. distribute-neg-out N/A

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

    6. mul-1-neg N/A

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

    7. sub-neg N/A

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

    8. mul-1-neg N/A

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

    9. lower-neg.f64 N/A

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

    10. mul-1-neg N/A

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

    11. sub-neg N/A

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

    12. mul-1-neg N/A

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

    13. distribute-neg-out N/A

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

    14. remove-double-neg N/A

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

    15. lower-fma.f64 56.3

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

11. Simplified 56.3%

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

Alternative 10: 43.7% 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 88.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 45.3

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

5. Simplified 45.3%

   \[\leadsto \color{blue}{-t} \]
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 2024207 
(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))