Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 89.0% → 99.6%

Time: 14.4s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * 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 - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * 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 - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

4. Applied rewrites 99.7%

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

Alternative 2: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - z, y, x \cdot \log y\right) - t\\ \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.98:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma (- 1.0 z) y (* x (log y))) t)))
   (if (<= (- x 1.0) -2e+16)
     t_1
     (if (<= (- x 1.0) -0.98) (- (fma (- 1.0 z) y (- (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((1.0 - z), y, (x * log(y))) - t;
	double tmp;
	if ((x - 1.0) <= -2e+16) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.98) {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(1.0 - z), y, Float64(x * log(y))) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -2e+16)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.98)
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(1.0 - z), $MachinePrecision] * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -2e+16], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.98], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e16 or -0.97999999999999998 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.4%

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

   if -2e16 < (-.f64 x #s(literal 1 binary64)) < -0.97999999999999998

   1. Initial program 84.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(1 - z, y, -\log y\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, x \cdot \log y\right) - t\\ \mathbf{elif}\;x - 1 \leq -0.98:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, x \cdot \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -1.0000002:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{elif}\;x - 1 \leq -0.999999999995:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x 1.0) -1.0000002)
   (- (* (- x 1.0) (log y)) t)
   (if (<= (- x 1.0) -0.999999999995)
     (- (fma (- 1.0 z) y (- (log y))) t)
     (- (fma (log y) (- x 1.0) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - 1.0) <= -1.0000002) {
		tmp = ((x - 1.0) * log(y)) - t;
	} else if ((x - 1.0) <= -0.999999999995) {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	} else {
		tmp = fma(log(y), (x - 1.0), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1.0000002)
		tmp = Float64(Float64(Float64(x - 1.0) * log(y)) - t);
	elseif (Float64(x - 1.0) <= -0.999999999995)
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	else
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.0000002], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.999999999995], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.00000019999999989

   1. Initial program 97.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if -1.00000019999999989 < (-.f64 x #s(literal 1 binary64)) < -0.999999999995

   1. Initial program 83.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

   if -0.999999999995 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.8%

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

Alternative 4: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x - 1 \leq -1.0000002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.5:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= (- x 1.0) -1.0000002)
     t_1
     (if (<= (- x 1.0) -0.5) (- (* z (log1p (- y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if ((x - 1.0) <= -1.0000002) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.5) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if ((x - 1.0) <= -1.0000002) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.5) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if (x - 1.0) <= -1.0000002:
		tmp = t_1
	elif (x - 1.0) <= -0.5:
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1.0000002)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.5)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.0000002], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.5], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.00000019999999989 or -0.5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if -1.00000019999999989 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 83.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 65.7

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

   5. Applied rewrites 65.7%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1.0000002:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x - 1 \leq -0.5:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 5e+211)
   (- (fma (log y) (- x 1.0) y) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 5e+211) {
		tmp = fma(log(y), (x - 1.0), y) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 5e+211)
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 5e+211], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 4.9999999999999995e211

   1. Initial program 92.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.4%

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

   if 4.9999999999999995e211 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 49.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 69.5

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

   5. Applied rewrites 69.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.5:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -2.15e-7) t_1 (if (<= x 9.5) (- (* (- 1.0 z) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_1;
	} else if (x <= 9.5) {
		tmp = ((1.0 - z) * y) - 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 = (x * log(y)) - t
    if (x <= (-2.15d-7)) then
        tmp = t_1
    else if (x <= 9.5d0) then
        tmp = ((1.0d0 - z) * y) - 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 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_1;
	} else if (x <= 9.5) {
		tmp = ((1.0 - z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -2.15e-7:
		tmp = t_1
	elif x <= 9.5:
		tmp = ((1.0 - z) * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = t_1;
	elseif (x <= 9.5)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - t;
	tmp = 0.0;
	if (x <= -2.15e-7)
		tmp = t_1;
	elseif (x <= 9.5)
		tmp = ((1.0 - z) * y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], t$95$1, If[LessEqual[x, 9.5], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e-7 or 9.5 < x

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if -2.1500000000000001e-7 < x < 9.5

   1. Initial program 83.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.6%

      \[\leadsto \left(1 - z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 9.5:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 5e+211)
   (- (* (- x 1.0) (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 5e+211) {
		tmp = ((x - 1.0) * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 5e+211) {
		tmp = ((x - 1.0) * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z - 1.0) <= 5e+211:
		tmp = ((x - 1.0) * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 5e+211)
		tmp = Float64(Float64(Float64(x - 1.0) * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 5e+211], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 4.9999999999999995e211

   1. Initial program 92.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 4.9999999999999995e211 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 49.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 69.5

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

   5. Applied rewrites 69.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. associate--r+ N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 9: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+30}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.6e+16) t_1 (if (<= x 4.9e+30) (- (* (- 1.0 z) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.6e+16) {
		tmp = t_1;
	} else if (x <= 4.9e+30) {
		tmp = ((1.0 - z) * y) - 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 = x * log(y)
    if (x <= (-1.6d+16)) then
        tmp = t_1
    else if (x <= 4.9d+30) then
        tmp = ((1.0d0 - z) * y) - 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 = x * Math.log(y);
	double tmp;
	if (x <= -1.6e+16) {
		tmp = t_1;
	} else if (x <= 4.9e+30) {
		tmp = ((1.0 - z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -1.6e+16:
		tmp = t_1
	elif x <= 4.9e+30:
		tmp = ((1.0 - z) * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.6e+16)
		tmp = t_1;
	elseif (x <= 4.9e+30)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -1.6e+16)
		tmp = t_1;
	elseif (x <= 4.9e+30)
		tmp = ((1.0 - z) * y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e+16], t$95$1, If[LessEqual[x, 4.9e+30], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e16 or 4.89999999999999984e30 < x

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -1.6e16 < x < 4.89999999999999984e30

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.6%

      \[\leadsto \left(1 - z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+30}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-36}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 0.145:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.06e-36) (- t) (if (<= t 0.145) (* (- y) z) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.06e-36) {
		tmp = -t;
	} else if (t <= 0.145) {
		tmp = -y * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.06d-36)) then
        tmp = -t
    else if (t <= 0.145d0) then
        tmp = -y * z
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.06e-36) {
		tmp = -t;
	} else if (t <= 0.145) {
		tmp = -y * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.06e-36:
		tmp = -t
	elif t <= 0.145:
		tmp = -y * z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.06e-36)
		tmp = Float64(-t);
	elseif (t <= 0.145)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.06e-36)
		tmp = -t;
	elseif (t <= 0.145)
		tmp = -y * z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.06e-36], (-t), If[LessEqual[t, 0.145], N[((-y) * z), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05999999999999999e-36 or 0.14499999999999999 < t

   1. Initial program 94.7%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   if -1.05999999999999999e-36 < t < 0.14499999999999999

   1. Initial program 83.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 19.6

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

   7. Applied rewrites 19.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 18.9%

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

Alternative 11: 46.1% accurate, 18.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((1.0 - z) * 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 = ((1.0d0 - z) * y) - t
end function

Java

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

Python

def code(x, y, z, t):
	return ((1.0 - z) * y) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. associate--r+ N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 49.5%

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

Alternative 12: 45.9% accurate, 20.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (-y * z) - 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 = (-y * z) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 49.2%

   \[\leadsto \left(-y\right) \cdot z - t \]
9. Add Preprocessing

Alternative 13: 35.3% accurate, 75.3× 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 89.3%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 38.7

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

5. Applied rewrites 38.7%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))