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

Percentage Accurate: 88.9% → 99.8%

Time: 21.5s

Alternatives: 15

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: 88.9% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. sub-neg 87.8%

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

   2. +-commutative 87.8%

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

   3. associate-+l+ 87.8%

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

   4. fma-def 87.8%

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

   5. sub-neg 87.8%

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

   6. metadata-eval 87.8%

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

   7. sub-neg 87.8%

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

   8. log1p-def 99.8%

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

   9. sub-neg 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (log1p (- y)) (+ z -1.0)) (* (log y) (+ -1.0 x))) t))

C

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

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log1p(-y) * (z + -1.0)) + (Math.log(y) * (-1.0 + x))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log1p(-y) * (z + -1.0)) + (math.log(y) * (-1.0 + x))) - t

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. *-commutative 87.8%

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

   2. sub-neg 87.8%

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

   3. metadata-eval 87.8%

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

   4. flip-+ 63.0%

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

   5. associate-*r/ 61.9%

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

   6. metadata-eval 61.9%

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

   7. fma-neg 61.9%

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

   8. metadata-eval 61.9%

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

   9. sub-neg 61.9%

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

   10. metadata-eval 61.9%

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

   11. +-commutative 61.9%

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

3. Applied egg-rr 61.9%

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

   1. associate-/l* 63.0%

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

   2. associate-/r/ 63.0%

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

5. Simplified 63.0%

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

6. Taylor expanded in x around 0 87.8%

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

   1. associate-+r+ 87.8%

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

   2. distribute-rgt-in 87.8%

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

   3. sub-neg 87.8%

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

   4. metadata-eval 87.8%

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

   5. sub-neg 87.8%

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

   6. log1p-def 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \left(\mathsf{log1p}\left(-y\right) \cdot \left(z + -1\right) + \log y \cdot \left(-1 + x\right)\right) - t \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in y around 0 98.9%

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

   1. mul-1-neg 98.9%

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

4. Simplified 98.9%

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

   1. +-commutative 98.9%

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

   2. fma-def 99.0%

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

   3. sub-neg 99.0%

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

   4. metadata-eval 99.0%

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

   5. +-commutative 99.0%

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

   6. *-commutative 99.0%

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

   7. sub-neg 99.0%

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

   8. metadata-eval 99.0%

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

   9. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in y around 0 99.7%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \left(-0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right) + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)\right)\right)} - t \]
3. Step-by-step derivation

   1. associate-+r+ 99.7%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right) + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)\right)} - t \]

   2. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(\left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right) + \left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right)} - t \]

   3. fma-def 99.7%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(z - 1\right), \log y \cdot \left(x - 1\right)\right)} + \left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   4. sub-neg 99.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \color{blue}{\left(z + \left(-1\right)\right)}, \log y \cdot \left(x - 1\right)\right) + \left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   5. metadata-eval 99.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(z + \color{blue}{-1}\right), \log y \cdot \left(x - 1\right)\right) + \left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   6. +-commutative 99.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \color{blue}{\left(-1 + z\right)}, \log y \cdot \left(x - 1\right)\right) + \left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   7. sub-neg 99.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \color{blue}{\left(x + \left(-1\right)\right)}\right) + \left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   8. metadata-eval 99.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \left(x + \color{blue}{-1}\right)\right) + \left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   9. +-commutative 99.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \color{blue}{\left(-1 + x\right)}\right) + \left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   10. sub-neg 99.7%

       \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \left(-1 + x\right)\right) + \left(-1 \cdot \left(y \cdot \color{blue}{\left(z + \left(-1\right)\right)}\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   11. metadata-eval 99.7%

       \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \left(-1 + x\right)\right) + \left(-1 \cdot \left(y \cdot \left(z + \color{blue}{-1}\right)\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   12. associate-*r* 99.7%

       \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \left(-1 + x\right)\right) + \left(\color{blue}{\left(-1 \cdot y\right) \cdot \left(z + -1\right)} + -0.5 \cdot \left({y}^{2} \cdot \left(z - 1\right)\right)\right)\right) - t \]

   13. sub-neg 99.7%

       \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \left(-1 + x\right)\right) + \left(\left(-1 \cdot y\right) \cdot \left(z + -1\right) + -0.5 \cdot \left({y}^{2} \cdot \color{blue}{\left(z + \left(-1\right)\right)}\right)\right)\right) - t \]

   14. metadata-eval 99.7%

       \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \left(-1 + x\right)\right) + \left(\left(-1 \cdot y\right) \cdot \left(z + -1\right) + -0.5 \cdot \left({y}^{2} \cdot \left(z + \color{blue}{-1}\right)\right)\right)\right) - t \]

4. Simplified 99.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, {y}^{3} \cdot \left(-1 + z\right), \log y \cdot \left(-1 + x\right)\right) + \left(-1 + z\right) \cdot \mathsf{fma}\left(-0.5, {y}^{2}, -y\right)\right)} - t \]

5. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. sub-neg 98.9%

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

   3. metadata-eval 98.9%

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

   4. +-commutative 98.9%

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

   5. fma-udef 99.0%

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

   6. associate-*r* 99.0%

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

   7. neg-mul-1 99.0%

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

   8. sub-neg 99.0%

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

   9. metadata-eval 99.0%

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

   10. +-commutative 99.0%

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

   11. distribute-rgt-in 99.0%

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

   12. neg-mul-1 99.0%

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

   13. remove-double-neg 99.0%

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

   14. distribute-rgt-neg-in 99.0%

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

   15. *-commutative 99.0%

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

   16. unsub-neg 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 5: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000 \lor \neg \left(-1 + x \leq -0.99999999995\right):\\ \;\;\;\;\left(x \cdot \log y - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -5000.0) (not (<= (+ -1.0 x) -0.99999999995)))
   (- (- (* x (log y)) (* y (+ z -1.0))) t)
   (- (- (- y (* z y)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -5000.0) || !((-1.0 + x) <= -0.99999999995)) {
		tmp = ((x * log(y)) - (y * (z + -1.0))) - t;
	} else {
		tmp = ((y - (z * y)) - log(y)) - 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 ((((-1.0d0) + x) <= (-5000.0d0)) .or. (.not. (((-1.0d0) + x) <= (-0.99999999995d0)))) then
        tmp = ((x * log(y)) - (y * (z + (-1.0d0)))) - t
    else
        tmp = ((y - (z * y)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -5000.0) || !((-1.0 + x) <= -0.99999999995)) {
		tmp = ((x * Math.log(y)) - (y * (z + -1.0))) - t;
	} else {
		tmp = ((y - (z * y)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -5000.0) or not ((-1.0 + x) <= -0.99999999995):
		tmp = ((x * math.log(y)) - (y * (z + -1.0))) - t
	else:
		tmp = ((y - (z * y)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -5000.0) || !(Float64(-1.0 + x) <= -0.99999999995))
		tmp = Float64(Float64(Float64(x * log(y)) - Float64(y * Float64(z + -1.0))) - t);
	else
		tmp = Float64(Float64(Float64(y - Float64(z * y)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -5000.0) || ~(((-1.0 + x) <= -0.99999999995)))
		tmp = ((x * log(y)) - (y * (z + -1.0))) - t;
	else
		tmp = ((y - (z * y)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -5000.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.99999999995]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -5e3 or -0.99999999995 < (-.f64 x 1)

   1. Initial program 92.1%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in x around inf 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   if -5e3 < (-.f64 x 1) < -0.99999999995

   1. Initial program 82.9%

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

   2. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. mul-1-neg 98.1%

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

      3. unsub-neg 98.1%

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

      4. associate-*r* 98.1%

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

      5. neg-mul-1 98.1%

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

      6. sub-neg 98.1%

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

      7. metadata-eval 98.1%

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

      8. +-commutative 98.1%

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

      9. distribute-rgt-in 98.1%

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

      10. neg-mul-1 98.1%

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

      11. remove-double-neg 98.1%

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

      12. distribute-rgt-neg-in 98.1%

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

      13. *-commutative 98.1%

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

      14. unsub-neg 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000 \lor \neg \left(-1 + x \leq -0.99999999995\right):\\ \;\;\;\;\left(x \cdot \log y - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \]

Alternative 6: 94.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -0.5:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -5e+71)
   (- (* x (log y)) t)
   (if (<= (+ -1.0 x) -0.5)
     (- (- (- y (* z y)) (log y)) t)
     (- (* (log y) (+ -1.0 x)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5e+71) {
		tmp = (x * log(y)) - t;
	} else if ((-1.0 + x) <= -0.5) {
		tmp = ((y - (z * y)) - log(y)) - t;
	} else {
		tmp = (log(y) * (-1.0 + x)) - 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 (((-1.0d0) + x) <= (-5d+71)) then
        tmp = (x * log(y)) - t
    else if (((-1.0d0) + x) <= (-0.5d0)) then
        tmp = ((y - (z * y)) - log(y)) - t
    else
        tmp = (log(y) * ((-1.0d0) + x)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5e+71) {
		tmp = (x * Math.log(y)) - t;
	} else if ((-1.0 + x) <= -0.5) {
		tmp = ((y - (z * y)) - Math.log(y)) - t;
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -5e+71:
		tmp = (x * math.log(y)) - t
	elif (-1.0 + x) <= -0.5:
		tmp = ((y - (z * y)) - math.log(y)) - t
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -5e+71)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(-1.0 + x) <= -0.5)
		tmp = Float64(Float64(Float64(y - Float64(z * y)) - log(y)) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -5e+71)
		tmp = (x * log(y)) - t;
	elseif ((-1.0 + x) <= -0.5)
		tmp = ((y - (z * y)) - log(y)) - t;
	else
		tmp = (log(y) * (-1.0 + x)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -5e+71], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.5], N[(N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -4.99999999999999972e71

   1. Initial program 95.6%

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

      1. *-commutative 95.6%

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

      2. sub-neg 95.6%

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

      3. metadata-eval 95.6%

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

      4. flip-+ 24.5%

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

      5. associate-*r/ 22.7%

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

      6. metadata-eval 22.7%

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

      7. fma-neg 22.7%

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

      8. metadata-eval 22.7%

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

      9. sub-neg 22.7%

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

      10. metadata-eval 22.7%

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

      11. +-commutative 22.7%

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

   3. Applied egg-rr 22.7%

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

      1. associate-/l* 24.4%

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

      2. associate-/r/ 24.4%

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

   5. Simplified 24.4%

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

   6. Taylor expanded in x around 0 95.6%

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

      1. associate-+r+ 95.6%

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

      2. distribute-rgt-in 95.6%

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

      3. sub-neg 95.6%

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

      4. metadata-eval 95.6%

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

      5. sub-neg 95.6%

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

      6. log1p-def 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around inf 95.1%

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

       1. *-commutative 95.1%

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

   11. Simplified 95.1%

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

   if -4.99999999999999972e71 < (-.f64 x 1) < -0.5

   1. Initial program 81.3%

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

   2. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in x around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. mul-1-neg 96.4%

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

      3. unsub-neg 96.4%

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

      4. associate-*r* 96.4%

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

      5. neg-mul-1 96.4%

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

      6. sub-neg 96.4%

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

      7. metadata-eval 96.4%

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

      8. +-commutative 96.4%

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

      9. distribute-rgt-in 96.4%

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

      10. neg-mul-1 96.4%

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

      11. remove-double-neg 96.4%

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

      12. distribute-rgt-neg-in 96.4%

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

      13. *-commutative 96.4%

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

      14. unsub-neg 96.4%

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

   7. Simplified 96.4%

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

   if -0.5 < (-.f64 x 1)

   1. Initial program 94.9%

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

   2. Taylor expanded in y around 0 94.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -0.5:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]

Alternative 7: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+253} \lor \neg \left(z + -1 \leq 5 \cdot 10^{+231}\right):\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ z -1.0) -2e+253) (not (<= (+ z -1.0) 5e+231)))
   (- (* z (log1p (- y))) t)
   (- (* (log y) (+ -1.0 x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z + -1.0) <= -2e+253) || !((z + -1.0) <= 5e+231)) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = (log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z + -1.0) <= -2e+253) || !((z + -1.0) <= 5e+231)) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z + -1.0) <= -2e+253) or not ((z + -1.0) <= 5e+231):
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z + -1.0) <= -2e+253) || !(Float64(z + -1.0) <= 5e+231))
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z + -1.0), $MachinePrecision], -2e+253], N[Not[LessEqual[N[(z + -1.0), $MachinePrecision], 5e+231]], $MachinePrecision]], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z 1) < -1.9999999999999999e253 or 5.00000000000000028e231 < (-.f64 z 1)

   1. Initial program 36.7%

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

   2. Taylor expanded in z around inf 29.1%

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

      1. *-commutative 29.1%

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

      2. sub-neg 29.1%

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

      3. mul-1-neg 29.1%

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

      4. log1p-def 86.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z - t \]

      5. mul-1-neg 86.1%

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

   4. Simplified 86.1%

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

   if -1.9999999999999999e253 < (-.f64 z 1) < 5.00000000000000028e231

   1. Initial program 93.1%

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

   2. Taylor expanded in y around 0 92.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+253} \lor \neg \left(z + -1 \leq 5 \cdot 10^{+231}\right):\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in y around 0 98.9%

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

   1. mul-1-neg 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 9: 60.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -1 \cdot 10^{+70}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;z + -1 \leq 10^{+113}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -1e+70)
   (- (fma y z t))
   (if (<= (+ z -1.0) 1e+113) (- (- (log y)) t) (- (- t) (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -1e+70) {
		tmp = -fma(y, z, t);
	} else if ((z + -1.0) <= 1e+113) {
		tmp = -log(y) - t;
	} else {
		tmp = -t - (z * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -1e+70)
		tmp = Float64(-fma(y, z, t));
	elseif (Float64(z + -1.0) <= 1e+113)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = Float64(Float64(-t) - Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -1e+70], (-N[(y * z + t), $MachinePrecision]), If[LessEqual[N[(z + -1.0), $MachinePrecision], 1e+113], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z 1) < -1.00000000000000007e70

   1. Initial program 77.4%

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

   2. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in z around inf 52.2%

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

      1. associate-*r* 52.2%

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

      2. neg-mul-1 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in y around 0 52.2%

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

      1. distribute-lft-out 52.2%

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

      2. +-commutative 52.2%

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

      3. fma-udef 52.2%

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

      4. neg-mul-1 52.2%

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

   10. Simplified 52.2%

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

   if -1.00000000000000007e70 < (-.f64 z 1) < 1e113

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around 0 60.9%

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

      1. +-commutative 60.9%

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

      2. mul-1-neg 60.9%

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

      3. unsub-neg 60.9%

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

      4. associate-*r* 60.9%

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

      5. neg-mul-1 60.9%

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

      6. sub-neg 60.9%

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

      7. metadata-eval 60.9%

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

      8. +-commutative 60.9%

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

      9. distribute-rgt-in 60.9%

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

      10. neg-mul-1 60.9%

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

      11. remove-double-neg 60.9%

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

      12. distribute-rgt-neg-in 60.9%

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

      13. *-commutative 60.9%

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

      14. unsub-neg 60.9%

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

   7. Simplified 60.9%

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

   8. Taylor expanded in y around 0 59.7%

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

      1. neg-mul-1 59.7%

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

   10. Simplified 59.7%

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

   if 1e113 < (-.f64 z 1)

   1. Initial program 60.9%

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

   2. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in z around inf 54.7%

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

      1. associate-*r* 54.7%

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

      2. neg-mul-1 54.7%

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

   7. Simplified 54.7%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -1 \cdot 10^{+70}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;z + -1 \leq 10^{+113}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \]

Alternative 10: 86.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-8} \lor \neg \left(x \leq 0.00072\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4e-8) (not (<= x 0.00072)))
   (- (* x (log y)) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4e-8) || !(x <= 0.00072)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -log(y) - 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 ((x <= (-4d-8)) .or. (.not. (x <= 0.00072d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4e-8) || !(x <= 0.00072)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4e-8) or not (x <= 0.00072):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4e-8) || !(x <= 0.00072))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4e-8) || ~((x <= 0.00072)))
		tmp = (x * log(y)) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4e-8], N[Not[LessEqual[x, 0.00072]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000001e-8 or 7.20000000000000045e-4 < x

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      2. sub-neg 92.2%

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

      3. metadata-eval 92.2%

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

      4. flip-+ 45.3%

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

      5. associate-*r/ 43.2%

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

      6. metadata-eval 43.2%

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

      7. fma-neg 43.2%

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

      8. metadata-eval 43.2%

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

      9. sub-neg 43.2%

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

      10. metadata-eval 43.2%

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

      11. +-commutative 43.2%

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

   3. Applied egg-rr 43.2%

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

      1. associate-/l* 45.3%

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

      2. associate-/r/ 45.3%

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

   5. Simplified 45.3%

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

   6. Taylor expanded in x around 0 92.2%

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

      1. associate-+r+ 92.2%

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

      2. distribute-rgt-in 92.2%

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

      3. sub-neg 92.2%

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

      4. metadata-eval 92.2%

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

      5. sub-neg 92.2%

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

      6. log1p-def 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 91.1%

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

       1. *-commutative 91.1%

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

   11. Simplified 91.1%

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

   if -4.0000000000000001e-8 < x < 7.20000000000000045e-4

   1. Initial program 82.8%

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

   2. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. mul-1-neg 98.2%

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

      3. unsub-neg 98.2%

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

      4. associate-*r* 98.2%

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

      5. neg-mul-1 98.2%

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

      6. sub-neg 98.2%

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

      7. metadata-eval 98.2%

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

      8. +-commutative 98.2%

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

      9. distribute-rgt-in 98.2%

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

      10. neg-mul-1 98.2%

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

      11. remove-double-neg 98.2%

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

      12. distribute-rgt-neg-in 98.2%

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

      13. *-commutative 98.2%

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

      14. unsub-neg 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y around 0 80.0%

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

      1. neg-mul-1 80.0%

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

   10. Simplified 80.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-8} \lor \neg \left(x \leq 0.00072\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]

Alternative 11: 87.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-8} \lor \neg \left(x \leq 0.00072\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4e-8) (not (<= x 0.00072)))
   (- (* x (log y)) t)
   (- (- y (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4e-8) || !(x <= 0.00072)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y - log(y)) - 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 ((x <= (-4d-8)) .or. (.not. (x <= 0.00072d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4e-8) || !(x <= 0.00072)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4e-8) or not (x <= 0.00072):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4e-8) || !(x <= 0.00072))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4e-8) || ~((x <= 0.00072)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4e-8], N[Not[LessEqual[x, 0.00072]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000001e-8 or 7.20000000000000045e-4 < x

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      2. sub-neg 92.2%

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

      3. metadata-eval 92.2%

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

      4. flip-+ 45.3%

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

      5. associate-*r/ 43.2%

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

      6. metadata-eval 43.2%

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

      7. fma-neg 43.2%

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

      8. metadata-eval 43.2%

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

      9. sub-neg 43.2%

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

      10. metadata-eval 43.2%

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

      11. +-commutative 43.2%

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

   3. Applied egg-rr 43.2%

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

      1. associate-/l* 45.3%

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

      2. associate-/r/ 45.3%

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

   5. Simplified 45.3%

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

   6. Taylor expanded in x around 0 92.2%

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

      1. associate-+r+ 92.2%

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

      2. distribute-rgt-in 92.2%

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

      3. sub-neg 92.2%

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

      4. metadata-eval 92.2%

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

      5. sub-neg 92.2%

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

      6. log1p-def 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 91.1%

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

       1. *-commutative 91.1%

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

   11. Simplified 91.1%

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

   if -4.0000000000000001e-8 < x < 7.20000000000000045e-4

   1. Initial program 82.8%

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

   2. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. mul-1-neg 98.2%

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

      3. unsub-neg 98.2%

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

      4. associate-*r* 98.2%

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

      5. neg-mul-1 98.2%

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

      6. sub-neg 98.2%

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

      7. metadata-eval 98.2%

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

      8. +-commutative 98.2%

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

      9. distribute-rgt-in 98.2%

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

      10. neg-mul-1 98.2%

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

      11. remove-double-neg 98.2%

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

      12. distribute-rgt-neg-in 98.2%

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

      13. *-commutative 98.2%

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

      14. unsub-neg 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in z around 0 80.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-8} \lor \neg \left(x \leq 0.00072\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]

Alternative 12: 42.7% accurate, 26.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-24} \lor \neg \left(t \leq 16000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2e-24) (not (<= t 16000000000.0))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2e-24) || !(t <= 16000000000.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	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 <= (-2d-24)) .or. (.not. (t <= 16000000000.0d0))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2e-24) || !(t <= 16000000000.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2e-24) or not (t <= 16000000000.0):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2e-24) || !(t <= 16000000000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2e-24) || ~((t <= 16000000000.0)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2e-24], N[Not[LessEqual[t, 16000000000.0]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.99999999999999985e-24 or 1.6e10 < t

   1. Initial program 96.7%

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

   2. Taylor expanded in t around inf 65.4%

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

      1. mul-1-neg 65.4%

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

   4. Simplified 65.4%

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

   if -1.99999999999999985e-24 < t < 1.6e10

   1. Initial program 80.5%

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

   2. Taylor expanded in y around 0 98.4%

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

      1. mul-1-neg 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in z around inf 20.8%

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

      1. associate-*r* 20.8%

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

      2. neg-mul-1 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in y around inf 20.9%

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

      1. associate-*r* 20.9%

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

      2. mul-1-neg 20.9%

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

   10. Simplified 20.9%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-24} \lor \neg \left(t \leq 16000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 13: 46.2% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in y around 0 98.9%

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

   1. mul-1-neg 98.9%

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

4. Simplified 98.9%

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

5. Taylor expanded in x around 0 61.5%

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

   1. +-commutative 61.5%

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

   2. mul-1-neg 61.5%

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

   3. unsub-neg 61.5%

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

   4. associate-*r* 61.5%

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

   5. neg-mul-1 61.5%

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

   6. sub-neg 61.5%

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

   7. metadata-eval 61.5%

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

   8. +-commutative 61.5%

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

   9. distribute-rgt-in 61.5%

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

   10. neg-mul-1 61.5%

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

   11. remove-double-neg 61.5%

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

   12. distribute-rgt-neg-in 61.5%

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

   13. *-commutative 61.5%

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

   14. unsub-neg 61.5%

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

7. Simplified 61.5%

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

8. Taylor expanded in y around inf 42.5%

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

9. Final simplification 42.5%

   \[\leadsto y \cdot \left(1 - z\right) - t \]

Alternative 14: 46.0% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

2. Taylor expanded in y around 0 98.9%

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

   1. mul-1-neg 98.9%

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

4. Simplified 98.9%

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

5. Taylor expanded in z around inf 42.3%

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

   1. associate-*r* 42.3%

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

   2. neg-mul-1 42.3%

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

7. Simplified 42.3%

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

8. Final simplification 42.3%

   \[\leadsto \left(-t\right) - z \cdot y \]

Alternative 15: 35.4% accurate, 107.5× 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 87.8%

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

2. Taylor expanded in t around inf 30.8%

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

   1. mul-1-neg 30.8%

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

4. Simplified 30.8%

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

5. Final simplification 30.8%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023308 
(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))