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

Percentage Accurate: 84.6% → 99.6%

Time: 12.0s

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: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 80.9

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

4. Applied rewrites 99.6%

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

Alternative 2: 87.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x \cdot \log y\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.8e+160)
   (fma (- y) z (* x (log y)))
   (if (<= z 1.06e+170) (fma (log y) x (- t)) (fma (log1p (- y)) z (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.8e+160) {
		tmp = fma(-y, z, (x * log(y)));
	} else if (z <= 1.06e+170) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = fma(log1p(-y), z, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.8e+160)
		tmp = fma(Float64(-y), z, Float64(x * log(y)));
	elseif (z <= 1.06e+170)
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = fma(log1p(Float64(-y)), z, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.8e+160], N[((-y) * z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.06e+170], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[Log[1 + (-y)], $MachinePrecision] * z + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8000000000000003e160

   1. Initial program 40.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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 89.3%

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

   if -4.8000000000000003e160 < z < 1.05999999999999998e170

   1. Initial program 95.9%

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

   3. Taylor expanded in 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 95.7

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

   5. Applied rewrites 95.7%

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

   if 1.05999999999999998e170 < z

   1. Initial program 44.6%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-neg.f64 84.6

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

   5. Applied rewrites 84.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x \cdot \log y\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \mathbf{if}\;z \leq -2.85 \cdot 10^{+163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log1p (- y)) z (- t))))
   (if (<= z -2.85e+163) t_1 (if (<= z 1.06e+170) (fma (log y) x (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log1p(-y), z, -t);
	double tmp;
	if (z <= -2.85e+163) {
		tmp = t_1;
	} else if (z <= 1.06e+170) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log1p(Float64(-y)), z, Float64(-t))
	tmp = 0.0
	if (z <= -2.85e+163)
		tmp = t_1;
	elseif (z <= 1.06e+170)
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[1 + (-y)], $MachinePrecision] * z + (-t)), $MachinePrecision]}, If[LessEqual[z, -2.85e+163], t$95$1, If[LessEqual[z, 1.06e+170], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8499999999999999e163 or 1.05999999999999998e170 < z

   1. Initial program 41.9%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-neg.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if -2.8499999999999999e163 < z < 1.05999999999999998e170

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \log y - t} \]
   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 95.2

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

   5. Applied rewrites 95.2%

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

Alternative 4: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\mathsf{fma}\left(z, y, t\right)\\ \mathbf{if}\;z \leq -2.85 \cdot 10^{+163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma z y t))))
   (if (<= z -2.85e+163) t_1 (if (<= z 1.06e+170) (fma (log y) x (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -fma(z, y, t);
	double tmp;
	if (z <= -2.85e+163) {
		tmp = t_1;
	} else if (z <= 1.06e+170) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(-fma(z, y, t))
	tmp = 0.0
	if (z <= -2.85e+163)
		tmp = t_1;
	elseif (z <= 1.06e+170)
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = (-N[(z * y + t), $MachinePrecision])}, If[LessEqual[z, -2.85e+163], t$95$1, If[LessEqual[z, 1.06e+170], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8499999999999999e163 or 1.05999999999999998e170 < z

   1. Initial program 41.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   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 77.0%

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

   if -2.8499999999999999e163 < z < 1.05999999999999998e170

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \log y - t} \]
   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 95.2

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

   5. Applied rewrites 95.2%

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

Alternative 5: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, t\right)\\ \mathbf{if}\;x \leq -3.35 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+120}:\\ \;\;\;\;-\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x t)))
   (if (<= x -3.35e+18) t_1 (if (<= x 1.12e+120) (- (fma z 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 <= -3.35e+18) {
		tmp = t_1;
	} else if (x <= 1.12e+120) {
		tmp = -fma(z, y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, t)
	tmp = 0.0
	if (x <= -3.35e+18)
		tmp = t_1;
	elseif (x <= 1.12e+120)
		tmp = Float64(-fma(z, 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, -3.35e+18], t$95$1, If[LessEqual[x, 1.12e+120], (-N[(z * y + t), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.35e18 or 1.12000000000000005e120 < x

   1. Initial program 96.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(-t, t, {\left(\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \log y \cdot x\right)\right)}^{2}\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, t\right)\right)}} \]

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {t}^{2} + {x}^{2} \cdot {\log y}^{2}}{t + x \cdot \log y}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot {\log y}^{2} + -1 \cdot {t}^{2}}}{t + x \cdot \log y} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-log.f64 39.4

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

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 81.5%

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

   if -3.35e18 < x < 1.12000000000000005e120

   1. Initial program 73.6%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   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 77.6%

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

Alternative 6: 99.1% accurate, 1.9× speedup

Math

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

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 81.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. 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 99.4

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

5. Applied rewrites 99.4%

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

Alternative 7: 45.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+170}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- y))))
   (if (<= z -9.2e+154) t_1 (if (<= z 1.1e+170) (- t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * -y;
	double tmp;
	if (z <= -9.2e+154) {
		tmp = t_1;
	} else if (z <= 1.1e+170) {
		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 <= (-9.2d+154)) then
        tmp = t_1
    else if (z <= 1.1d+170) 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 <= -9.2e+154) {
		tmp = t_1;
	} else if (z <= 1.1e+170) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * -y
	tmp = 0
	if z <= -9.2e+154:
		tmp = t_1
	elif z <= 1.1e+170:
		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 <= -9.2e+154)
		tmp = t_1;
	elseif (z <= 1.1e+170)
		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 <= -9.2e+154)
		tmp = t_1;
	elseif (z <= 1.1e+170)
		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, -9.2e+154], t$95$1, If[LessEqual[z, 1.1e+170], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.1999999999999999e154 or 1.09999999999999994e170 < z

   1. Initial program 43.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 58.4%

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

   if -9.1999999999999999e154 < z < 1.09999999999999994e170

   1. Initial program 96.3%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 48.0

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

   5. Applied rewrites 48.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+154}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+170}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.4% 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 81.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. 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 99.4

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

5. Applied rewrites 99.4%

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

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 58.5%

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

Alternative 9: 42.4% accurate, 73.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.3

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

5. Applied rewrites 39.3%

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

Alternative 10: 2.3% accurate, 220.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z, t)
	return t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.3

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

5. Applied rewrites 39.3%

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

7. Applied rewrites 24.1%

   \[\leadsto \frac{\left(-t\right) \cdot t}{\color{blue}{0 + t}} \]
8. Step-by-step derivation

9. Applied rewrites 2.4%

   \[\leadsto t \]
10. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (*
   (- z)
   (+
    (+ (* 0.5 (* y y)) y)
    (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y)))))
  (- t (* x (log y)))))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * Math.log(y)));
}

Python

def code(x, y, z, t):
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * math.log(y)))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(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))