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

Percentage Accurate: 89.0% → 99.8%

Time: 12.6s

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: 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. associate--l+ 91.3%

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

   2. +-commutative 91.3%

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

   3. associate-+l- 91.3%

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

   4. *-commutative 91.3%

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

   5. *-commutative 91.3%

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

   6. sub-neg 91.3%

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

   7. metadata-eval 91.3%

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

   8. sub-neg 91.3%

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

   9. log1p-def 99.8%

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

   10. sub-neg 99.8%

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

   11. +-commutative 99.8%

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

   12. *-commutative 99.8%

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

   13. distribute-rgt-neg-in 99.8%

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

   14. fma-def 99.8%

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

   15. neg-sub0 99.8%

       \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{0 - \left(x - 1\right)}, t\right) \]

   16. associate--r- 99.8%

       \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{\left(0 - x\right) + 1}, t\right) \]

   17. neg-sub0 99.8%

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

   18. +-commutative 99.8%

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

   19. unsub-neg 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. associate--l+ 91.3%

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

   2. +-commutative 91.3%

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

   3. associate-+l- 91.3%

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

   4. *-commutative 91.3%

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

   5. *-commutative 91.3%

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

   6. sub-neg 91.3%

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

   7. metadata-eval 91.3%

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

   8. sub-neg 91.3%

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

   9. log1p-def 99.8%

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

   10. sub-neg 99.8%

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

   11. +-commutative 99.8%

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

   12. *-commutative 99.8%

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

   13. distribute-rgt-neg-in 99.8%

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

   14. fma-def 99.8%

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

   15. neg-sub0 99.8%

       \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{0 - \left(x - 1\right)}, t\right) \]

   16. associate--r- 99.8%

       \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{\left(0 - x\right) + 1}, t\right) \]

   17. neg-sub0 99.8%

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

   18. +-commutative 99.8%

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

   19. unsub-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 99.5%

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

   1. associate-*r* 99.5%

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

   2. *-commutative 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

   6. distribute-lft-in 99.5%

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

   7. metadata-eval 99.5%

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

   8. neg-mul-1 99.5%

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

   9. unsub-neg 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 95.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -500:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{elif}\;-1 + x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -500.0)
   (- (* (log y) (+ -1.0 x)) t)
   (if (<= (+ -1.0 x) 4e+15)
     (- (- (* y (- 1.0 z)) (log y)) t)
     (- (* (log y) x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -500.0) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else if ((-1.0 + x) <= 4e+15) {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	} else {
		tmp = (log(y) * 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) <= (-500.0d0)) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else if (((-1.0d0) + x) <= 4d+15) then
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    else
        tmp = (log(y) * 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) <= -500.0) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else if ((-1.0 + x) <= 4e+15) {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	} else {
		tmp = (Math.log(y) * x) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -500.0:
		tmp = (math.log(y) * (-1.0 + x)) - t
	elif (-1.0 + x) <= 4e+15:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	else:
		tmp = (math.log(y) * x) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -500.0)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	elseif (Float64(-1.0 + x) <= 4e+15)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	else
		tmp = Float64(Float64(log(y) * x) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -500.0)
		tmp = (log(y) * (-1.0 + x)) - t;
	elseif ((-1.0 + x) <= 4e+15)
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	else
		tmp = (log(y) * x) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -500.0], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 4e+15], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -500

   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. Taylor expanded in y around 0 94.0%

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

   if -500 < (-.f64 x 1) < 4e15

   1. Initial program 86.0%

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

   2. Taylor expanded in x around 0 85.3%

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

      1. mul-1-neg 85.3%

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

      2. unsub-neg 85.3%

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

      3. *-commutative 85.3%

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

      4. sub-neg 85.3%

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

      5. metadata-eval 85.3%

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

      6. sub-neg 85.3%

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

      7. mul-1-neg 85.3%

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

      8. log1p-def 99.2%

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

      9. mul-1-neg 99.2%

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

      10. +-commutative 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in y around 0 98.9%

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

      1. neg-mul-1 98.9%

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

      2. unsub-neg 98.9%

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

      3. associate-*r* 98.9%

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

      4. sub-neg 98.9%

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

      5. metadata-eval 98.9%

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

      6. +-commutative 98.9%

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

      7. distribute-lft-in 98.9%

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

      8. metadata-eval 98.9%

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

      9. neg-mul-1 98.9%

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

      10. sub-neg 98.9%

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

      11. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if 4e15 < (-.f64 x 1)

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0 99.6%

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

   3. Taylor expanded in x around inf 99.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -500:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{elif}\;-1 + x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]

Alternative 4: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\log y\\ t_2 := \log y \cdot x\\ t_3 := y \cdot \left(1 - z\right) - t\\ \mathbf{if}\;x \leq -65000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-216}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-140}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+21}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log y))) (t_2 (* (log y) x)) (t_3 (- (* y (- 1.0 z)) t)))
   (if (<= x -65000000000000.0)
     t_2
     (if (<= x -1.35e-216)
       t_3
       (if (<= x -1.9e-295)
         t_1
         (if (<= x 2.2e-140)
           t_3
           (if (<= x 1.8e-98) t_1 (if (<= x 1.8e+21) t_3 t_2))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -log(y);
	double t_2 = log(y) * x;
	double t_3 = (y * (1.0 - z)) - t;
	double tmp;
	if (x <= -65000000000000.0) {
		tmp = t_2;
	} else if (x <= -1.35e-216) {
		tmp = t_3;
	} else if (x <= -1.9e-295) {
		tmp = t_1;
	} else if (x <= 2.2e-140) {
		tmp = t_3;
	} else if (x <= 1.8e-98) {
		tmp = t_1;
	} else if (x <= 1.8e+21) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = -log(y)
    t_2 = log(y) * x
    t_3 = (y * (1.0d0 - z)) - t
    if (x <= (-65000000000000.0d0)) then
        tmp = t_2
    else if (x <= (-1.35d-216)) then
        tmp = t_3
    else if (x <= (-1.9d-295)) then
        tmp = t_1
    else if (x <= 2.2d-140) then
        tmp = t_3
    else if (x <= 1.8d-98) then
        tmp = t_1
    else if (x <= 1.8d+21) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -Math.log(y);
	double t_2 = Math.log(y) * x;
	double t_3 = (y * (1.0 - z)) - t;
	double tmp;
	if (x <= -65000000000000.0) {
		tmp = t_2;
	} else if (x <= -1.35e-216) {
		tmp = t_3;
	} else if (x <= -1.9e-295) {
		tmp = t_1;
	} else if (x <= 2.2e-140) {
		tmp = t_3;
	} else if (x <= 1.8e-98) {
		tmp = t_1;
	} else if (x <= 1.8e+21) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -math.log(y)
	t_2 = math.log(y) * x
	t_3 = (y * (1.0 - z)) - t
	tmp = 0
	if x <= -65000000000000.0:
		tmp = t_2
	elif x <= -1.35e-216:
		tmp = t_3
	elif x <= -1.9e-295:
		tmp = t_1
	elif x <= 2.2e-140:
		tmp = t_3
	elif x <= 1.8e-98:
		tmp = t_1
	elif x <= 1.8e+21:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-log(y))
	t_2 = Float64(log(y) * x)
	t_3 = Float64(Float64(y * Float64(1.0 - z)) - t)
	tmp = 0.0
	if (x <= -65000000000000.0)
		tmp = t_2;
	elseif (x <= -1.35e-216)
		tmp = t_3;
	elseif (x <= -1.9e-295)
		tmp = t_1;
	elseif (x <= 2.2e-140)
		tmp = t_3;
	elseif (x <= 1.8e-98)
		tmp = t_1;
	elseif (x <= 1.8e+21)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -log(y);
	t_2 = log(y) * x;
	t_3 = (y * (1.0 - z)) - t;
	tmp = 0.0;
	if (x <= -65000000000000.0)
		tmp = t_2;
	elseif (x <= -1.35e-216)
		tmp = t_3;
	elseif (x <= -1.9e-295)
		tmp = t_1;
	elseif (x <= 2.2e-140)
		tmp = t_3;
	elseif (x <= 1.8e-98)
		tmp = t_1;
	elseif (x <= 1.8e+21)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = (-N[Log[y], $MachinePrecision])}, Block[{t$95$2 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -65000000000000.0], t$95$2, If[LessEqual[x, -1.35e-216], t$95$3, If[LessEqual[x, -1.9e-295], t$95$1, If[LessEqual[x, 2.2e-140], t$95$3, If[LessEqual[x, 1.8e-98], t$95$1, If[LessEqual[x, 1.8e+21], t$95$3, t$95$2]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e13 or 1.8e21 < x

   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 y around 0 96.2%

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

   3. Taylor expanded in x around inf 79.6%

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

   if -6.5e13 < x < -1.35e-216 or -1.90000000000000009e-295 < x < 2.1999999999999999e-140 or 1.8000000000000001e-98 < x < 1.8e21

   1. Initial program 85.5%

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

      1. associate--l+ 85.5%

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

      2. +-commutative 85.5%

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

      3. associate-+l- 85.5%

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

      4. *-commutative 85.5%

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

      5. *-commutative 85.5%

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

      6. sub-neg 85.5%

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

      7. metadata-eval 85.5%

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

      8. sub-neg 85.5%

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

      9. log1p-def 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. fma-def 100.0%

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

      15. neg-sub0 100.0%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{0 - \left(x - 1\right)}, t\right) \]

      16. associate--r- 100.0%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{\left(0 - x\right) + 1}, t\right) \]

      17. neg-sub0 100.0%

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

      18. +-commutative 100.0%

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

      19. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-mul-1 100.0%

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

      9. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 70.1%

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

   if -1.35e-216 < x < -1.90000000000000009e-295 or 2.1999999999999999e-140 < x < 1.8000000000000001e-98

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0 91.4%

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

      1. mul-1-neg 91.4%

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

      2. unsub-neg 91.4%

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

      3. *-commutative 91.4%

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

      4. sub-neg 91.4%

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

      5. metadata-eval 91.4%

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

      6. sub-neg 91.4%

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

      7. mul-1-neg 91.4%

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

      8. log1p-def 100.0%

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

      9. mul-1-neg 100.0%

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

      10. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 88.6%

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

      1. neg-mul-1 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in t around 0 75.9%

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

      1. mul-1-neg 75.9%

         \[\leadsto \color{blue}{-\log y} \]

   10. Simplified 75.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -65000000000000:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-216}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-295}:\\ \;\;\;\;-\log y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-140}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-98}:\\ \;\;\;\;-\log y\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]

Alternative 5: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -4000:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{elif}\;-1 + x \leq 10^{+17}:\\ \;\;\;\;y - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -4000.0)
   (* (log y) (+ -1.0 x))
   (if (<= (+ -1.0 x) 1e+17) (- y (+ (log y) t)) (* (log y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -4000.0) {
		tmp = log(y) * (-1.0 + x);
	} else if ((-1.0 + x) <= 1e+17) {
		tmp = y - (log(y) + t);
	} else {
		tmp = log(y) * x;
	}
	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) <= (-4000.0d0)) then
        tmp = log(y) * ((-1.0d0) + x)
    else if (((-1.0d0) + x) <= 1d+17) then
        tmp = y - (log(y) + t)
    else
        tmp = log(y) * x
    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) <= -4000.0) {
		tmp = Math.log(y) * (-1.0 + x);
	} else if ((-1.0 + x) <= 1e+17) {
		tmp = y - (Math.log(y) + t);
	} else {
		tmp = Math.log(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -4000.0:
		tmp = math.log(y) * (-1.0 + x)
	elif (-1.0 + x) <= 1e+17:
		tmp = y - (math.log(y) + t)
	else:
		tmp = math.log(y) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -4000.0)
		tmp = Float64(log(y) * Float64(-1.0 + x));
	elseif (Float64(-1.0 + x) <= 1e+17)
		tmp = Float64(y - Float64(log(y) + t));
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -4000.0)
		tmp = log(y) * (-1.0 + x);
	elseif ((-1.0 + x) <= 1e+17)
		tmp = y - (log(y) + t);
	else
		tmp = log(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -4000.0], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 1e+17], N[(y - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -4e3

   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. Taylor expanded in y around 0 93.9%

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

   3. Taylor expanded in t around 0 79.1%

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

   if -4e3 < (-.f64 x 1) < 1e17

   1. Initial program 86.2%

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

   2. Taylor expanded in x around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. unsub-neg 85.1%

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

      3. *-commutative 85.1%

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

      4. sub-neg 85.1%

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

      5. metadata-eval 85.1%

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

      6. sub-neg 85.1%

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

      7. mul-1-neg 85.1%

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

      8. log1p-def 98.8%

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

      9. mul-1-neg 98.8%

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

      10. +-commutative 98.8%

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

   4. Simplified 98.8%

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

   5. Taylor expanded in z around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. sub-neg 85.1%

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

      3. log1p-def 85.1%

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

   7. Simplified 85.1%

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

   8. Taylor expanded in y around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. +-commutative 84.7%

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

      3. unsub-neg 84.7%

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

   10. Simplified 84.7%

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

   if 1e17 < (-.f64 x 1)

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0 99.6%

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

   3. Taylor expanded in x around inf 80.1%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -4000:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{elif}\;-1 + x \leq 10^{+17}:\\ \;\;\;\;y - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]

Alternative 6: 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 91.3%

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

2. Taylor expanded in y around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

   4. *-commutative 99.5%

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

   5. mul-1-neg 99.5%

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

   6. unsub-neg 99.5%

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

   7. *-commutative 99.5%

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

   8. sub-neg 99.5%

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

   9. metadata-eval 99.5%

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

   10. +-commutative 99.5%

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

4. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 7: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(-t\right) - \log y\\ \mathbf{if}\;x \leq -20000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-142}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- (- t) (log y))))
   (if (<= x -20000000000000.0)
     t_1
     (if (<= x 2.35e-172)
       t_2
       (if (<= x 4.1e-142)
         (- (* y (- 1.0 z)) t)
         (if (<= x 2.8e+19) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = -t - log(y);
	double tmp;
	if (x <= -20000000000000.0) {
		tmp = t_1;
	} else if (x <= 2.35e-172) {
		tmp = t_2;
	} else if (x <= 4.1e-142) {
		tmp = (y * (1.0 - z)) - t;
	} else if (x <= 2.8e+19) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = -t - log(y)
    if (x <= (-20000000000000.0d0)) then
        tmp = t_1
    else if (x <= 2.35d-172) then
        tmp = t_2
    else if (x <= 4.1d-142) then
        tmp = (y * (1.0d0 - z)) - t
    else if (x <= 2.8d+19) then
        tmp = t_2
    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 = Math.log(y) * x;
	double t_2 = -t - Math.log(y);
	double tmp;
	if (x <= -20000000000000.0) {
		tmp = t_1;
	} else if (x <= 2.35e-172) {
		tmp = t_2;
	} else if (x <= 4.1e-142) {
		tmp = (y * (1.0 - z)) - t;
	} else if (x <= 2.8e+19) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = -t - math.log(y)
	tmp = 0
	if x <= -20000000000000.0:
		tmp = t_1
	elif x <= 2.35e-172:
		tmp = t_2
	elif x <= 4.1e-142:
		tmp = (y * (1.0 - z)) - t
	elif x <= 2.8e+19:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(-t) - log(y))
	tmp = 0.0
	if (x <= -20000000000000.0)
		tmp = t_1;
	elseif (x <= 2.35e-172)
		tmp = t_2;
	elseif (x <= 4.1e-142)
		tmp = Float64(Float64(y * Float64(1.0 - z)) - t);
	elseif (x <= 2.8e+19)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = -t - log(y);
	tmp = 0.0;
	if (x <= -20000000000000.0)
		tmp = t_1;
	elseif (x <= 2.35e-172)
		tmp = t_2;
	elseif (x <= 4.1e-142)
		tmp = (y * (1.0 - z)) - t;
	elseif (x <= 2.8e+19)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -20000000000000.0], t$95$1, If[LessEqual[x, 2.35e-172], t$95$2, If[LessEqual[x, 4.1e-142], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 2.8e+19], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e13 or 2.8e19 < x

   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 y around 0 96.2%

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

   3. Taylor expanded in x around inf 79.6%

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

   if -2e13 < x < 2.34999999999999988e-172 or 4.1e-142 < x < 2.8e19

   1. Initial program 89.5%

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

   2. Taylor expanded in x around 0 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

      3. *-commutative 86.9%

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

      4. sub-neg 86.9%

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

      5. metadata-eval 86.9%

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

      6. sub-neg 86.9%

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

      7. mul-1-neg 86.9%

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

      8. log1p-def 97.3%

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

      9. mul-1-neg 97.3%

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

      10. +-commutative 97.3%

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

   4. Simplified 97.3%

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

   5. Taylor expanded in y around 0 86.4%

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

      1. neg-mul-1 86.4%

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

   7. Simplified 86.4%

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

   if 2.34999999999999988e-172 < x < 4.1e-142

   1. Initial program 52.9%

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

      1. associate--l+ 52.9%

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

      2. +-commutative 52.9%

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

      3. associate-+l- 52.9%

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

      4. *-commutative 52.9%

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

      5. *-commutative 52.9%

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

      6. sub-neg 52.9%

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

      7. metadata-eval 52.9%

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

      8. sub-neg 52.9%

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

      9. log1p-def 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. fma-def 100.0%

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

      15. neg-sub0 100.0%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{0 - \left(x - 1\right)}, t\right) \]

      16. associate--r- 100.0%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{\left(0 - x\right) + 1}, t\right) \]

      17. neg-sub0 100.0%

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

      18. +-commutative 100.0%

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

      19. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-mul-1 100.0%

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

      9. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around inf 100.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000000000:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-172}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-142}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+19}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]

Alternative 8: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -4000:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{elif}\;-1 + x \leq 10^{+17}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -4000.0)
   (* (log y) (+ -1.0 x))
   (if (<= (+ -1.0 x) 1e+17) (- (- t) (log y)) (* (log y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -4000.0) {
		tmp = log(y) * (-1.0 + x);
	} else if ((-1.0 + x) <= 1e+17) {
		tmp = -t - log(y);
	} else {
		tmp = log(y) * x;
	}
	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) <= (-4000.0d0)) then
        tmp = log(y) * ((-1.0d0) + x)
    else if (((-1.0d0) + x) <= 1d+17) then
        tmp = -t - log(y)
    else
        tmp = log(y) * x
    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) <= -4000.0) {
		tmp = Math.log(y) * (-1.0 + x);
	} else if ((-1.0 + x) <= 1e+17) {
		tmp = -t - Math.log(y);
	} else {
		tmp = Math.log(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -4000.0:
		tmp = math.log(y) * (-1.0 + x)
	elif (-1.0 + x) <= 1e+17:
		tmp = -t - math.log(y)
	else:
		tmp = math.log(y) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -4000.0)
		tmp = Float64(log(y) * Float64(-1.0 + x));
	elseif (Float64(-1.0 + x) <= 1e+17)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -4000.0)
		tmp = log(y) * (-1.0 + x);
	elseif ((-1.0 + x) <= 1e+17)
		tmp = -t - log(y);
	else
		tmp = log(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -4000.0], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 1e+17], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -4e3

   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. Taylor expanded in y around 0 93.9%

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

   3. Taylor expanded in t around 0 79.1%

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

   if -4e3 < (-.f64 x 1) < 1e17

   1. Initial program 86.2%

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

   2. Taylor expanded in x around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. unsub-neg 85.1%

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

      3. *-commutative 85.1%

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

      4. sub-neg 85.1%

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

      5. metadata-eval 85.1%

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

      6. sub-neg 85.1%

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

      7. mul-1-neg 85.1%

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

      8. log1p-def 98.8%

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

      9. mul-1-neg 98.8%

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

      10. +-commutative 98.8%

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

   4. Simplified 98.8%

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

   5. Taylor expanded in y around 0 84.6%

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

      1. neg-mul-1 84.6%

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

   7. Simplified 84.6%

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

   if 1e17 < (-.f64 x 1)

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0 99.6%

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

   3. Taylor expanded in x around inf 80.1%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -4000:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{elif}\;-1 + x \leq 10^{+17}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]

Alternative 9: 86.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -14000 \lor \neg \left(t \leq 320\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -14000.0) (not (<= t 320.0)))
   (- (* (log y) x) t)
   (* (log y) (+ -1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -14000.0) || !(t <= 320.0)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = log(y) * (-1.0 + x);
	}
	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 <= (-14000.0d0)) .or. (.not. (t <= 320.0d0))) then
        tmp = (log(y) * x) - t
    else
        tmp = log(y) * ((-1.0d0) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -14000.0) || !(t <= 320.0)) {
		tmp = (Math.log(y) * x) - t;
	} else {
		tmp = Math.log(y) * (-1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -14000.0) or not (t <= 320.0):
		tmp = (math.log(y) * x) - t
	else:
		tmp = math.log(y) * (-1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -14000.0) || !(t <= 320.0))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(log(y) * Float64(-1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -14000.0) || ~((t <= 320.0)))
		tmp = (log(y) * x) - t;
	else
		tmp = log(y) * (-1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -14000.0], N[Not[LessEqual[t, 320.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -14000 or 320 < t

   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. Taylor expanded in y around 0 94.8%

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

   3. Taylor expanded in x around inf 93.3%

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

   if -14000 < t < 320

   1. Initial program 88.0%

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

   2. Taylor expanded in y around 0 87.0%

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

   3. Taylor expanded in t around 0 87.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -14000 \lor \neg \left(t \leq 320\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \end{array} \]

Alternative 10: 87.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

2. Taylor expanded in y around 0 90.9%

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

3. Final simplification 90.9%

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

Alternative 11: 54.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-53} \lor \neg \left(t \leq 3.5 \cdot 10^{-33}\right):\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.05e-53) (not (<= t 3.5e-33)))
   (- (* y (- 1.0 z)) t)
   (- (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.05e-53) || !(t <= 3.5e-33)) {
		tmp = (y * (1.0 - z)) - t;
	} else {
		tmp = -log(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 <= (-1.05d-53)) .or. (.not. (t <= 3.5d-33))) then
        tmp = (y * (1.0d0 - z)) - t
    else
        tmp = -log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.05e-53) || !(t <= 3.5e-33)) {
		tmp = (y * (1.0 - z)) - t;
	} else {
		tmp = -Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.05e-53) or not (t <= 3.5e-33):
		tmp = (y * (1.0 - z)) - t
	else:
		tmp = -math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.05e-53) || !(t <= 3.5e-33))
		tmp = Float64(Float64(y * Float64(1.0 - z)) - t);
	else
		tmp = Float64(-log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.05e-53) || ~((t <= 3.5e-33)))
		tmp = (y * (1.0 - z)) - t;
	else
		tmp = -log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.05e-53], N[Not[LessEqual[t, 3.5e-33]], $MachinePrecision]], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[Log[y], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if t < -1.04999999999999989e-53 or 3.4999999999999999e-33 < t

   1. Initial program 93.3%

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

      1. associate--l+ 93.3%

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

      2. +-commutative 93.3%

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

      3. associate-+l- 93.3%

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

      4. *-commutative 93.3%

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

      5. *-commutative 93.3%

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

      6. sub-neg 93.3%

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

      7. metadata-eval 93.3%

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

      8. sub-neg 93.3%

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

      9. log1p-def 99.8%

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

      10. sub-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. *-commutative 99.8%

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

      13. distribute-rgt-neg-in 99.8%

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

      14. fma-def 99.8%

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

      15. neg-sub0 99.8%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{0 - \left(x - 1\right)}, t\right) \]

      16. associate--r- 99.8%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{\left(0 - x\right) + 1}, t\right) \]

      17. neg-sub0 99.8%

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

      18. +-commutative 99.8%

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

      19. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. neg-mul-1 99.8%

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

      9. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around inf 65.7%

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

   if -1.04999999999999989e-53 < t < 3.4999999999999999e-33

   1. Initial program 89.0%

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

   2. Taylor expanded in x around 0 43.6%

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

      1. mul-1-neg 43.6%

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

      2. unsub-neg 43.6%

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

      3. *-commutative 43.6%

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

      4. sub-neg 43.6%

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

      5. metadata-eval 43.6%

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

      6. sub-neg 43.6%

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

      7. mul-1-neg 43.6%

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

      8. log1p-def 54.3%

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

      9. mul-1-neg 54.3%

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

      10. +-commutative 54.3%

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

   4. Simplified 54.3%

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

   5. Taylor expanded in y around 0 42.6%

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

      1. neg-mul-1 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in t around 0 42.6%

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

      1. mul-1-neg 42.6%

         \[\leadsto \color{blue}{-\log y} \]

   10. Simplified 42.6%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-53} \lor \neg \left(t \leq 3.5 \cdot 10^{-33}\right):\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \]

Alternative 12: 43.5% accurate, 23.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-35}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 320:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e-35) (- t) (if (<= t 320.0) (* y (- 1.0 z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e-35) {
		tmp = -t;
	} else if (t <= 320.0) {
		tmp = y * (1.0 - 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 <= (-9d-35)) then
        tmp = -t
    else if (t <= 320.0d0) then
        tmp = y * (1.0d0 - 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 <= -9e-35) {
		tmp = -t;
	} else if (t <= 320.0) {
		tmp = y * (1.0 - z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e-35:
		tmp = -t
	elif t <= 320.0:
		tmp = y * (1.0 - z)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e-35)
		tmp = Float64(-t);
	elseif (t <= 320.0)
		tmp = Float64(y * Float64(1.0 - z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e-35)
		tmp = -t;
	elseif (t <= 320.0)
		tmp = y * (1.0 - z);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e-35], (-t), If[LessEqual[t, 320.0], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -9.0000000000000002e-35 or 320 < t

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 95.0%

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

   3. Taylor expanded in t around inf 62.5%

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

      1. neg-mul-1 62.5%

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

   5. Simplified 62.5%

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

   if -9.0000000000000002e-35 < t < 320

   1. Initial program 87.3%

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

      1. associate--l+ 87.3%

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

      2. +-commutative 87.3%

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

      3. associate-+l- 87.3%

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

      4. *-commutative 87.3%

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

      5. *-commutative 87.3%

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

      6. sub-neg 87.3%

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

      7. metadata-eval 87.3%

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

      8. sub-neg 87.3%

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

      9. log1p-def 99.8%

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

      10. sub-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. *-commutative 99.8%

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

      13. distribute-rgt-neg-in 99.8%

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

      14. fma-def 99.8%

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

      15. neg-sub0 99.8%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{0 - \left(x - 1\right)}, t\right) \]

      16. associate--r- 99.8%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{\left(0 - x\right) + 1}, t\right) \]

      17. neg-sub0 99.8%

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

      18. +-commutative 99.8%

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

      19. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.2%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

      5. +-commutative 99.2%

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

      6. distribute-lft-in 99.2%

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

      7. metadata-eval 99.2%

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

      8. neg-mul-1 99.2%

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

      9. unsub-neg 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around inf 62.0%

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

      1. associate-*r* 62.0%

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

      2. mul-1-neg 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in y around inf 16.2%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-35}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 320:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 13: 43.2% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.42 \cdot 10^{-34}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 320:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.42e-34) (- t) (if (<= t 320.0) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.42e-34) {
		tmp = -t;
	} else if (t <= 320.0) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.42d-34)) then
        tmp = -t
    else if (t <= 320.0d0) then
        tmp = z * -y
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.42e-34) {
		tmp = -t;
	} else if (t <= 320.0) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.42e-34:
		tmp = -t
	elif t <= 320.0:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.42e-34)
		tmp = Float64(-t);
	elseif (t <= 320.0)
		tmp = Float64(z * Float64(-y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.42e-34)
		tmp = -t;
	elseif (t <= 320.0)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.42e-34], (-t), If[LessEqual[t, 320.0], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.42000000000000003e-34 or 320 < t

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 95.0%

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

   3. Taylor expanded in t around inf 62.5%

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

      1. neg-mul-1 62.5%

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

   5. Simplified 62.5%

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

   if -1.42000000000000003e-34 < t < 320

   1. Initial program 87.3%

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

      1. associate--l+ 87.3%

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

      2. +-commutative 87.3%

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

      3. associate-+l- 87.3%

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

      4. *-commutative 87.3%

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

      5. *-commutative 87.3%

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

      6. sub-neg 87.3%

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

      7. metadata-eval 87.3%

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

      8. sub-neg 87.3%

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

      9. log1p-def 99.8%

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

      10. sub-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. *-commutative 99.8%

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

      13. distribute-rgt-neg-in 99.8%

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

      14. fma-def 99.8%

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

      15. neg-sub0 99.8%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{0 - \left(x - 1\right)}, t\right) \]

      16. associate--r- 99.8%

          \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{\left(0 - x\right) + 1}, t\right) \]

      17. neg-sub0 99.8%

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

      18. +-commutative 99.8%

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

      19. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.2%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

      5. +-commutative 99.2%

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

      6. distribute-lft-in 99.2%

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

      7. metadata-eval 99.2%

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

      8. neg-mul-1 99.2%

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

      9. unsub-neg 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around inf 62.0%

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

      1. associate-*r* 62.0%

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

      2. mul-1-neg 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in z around inf 15.6%

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

       1. mul-1-neg 15.6%

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

       2. distribute-rgt-neg-in 15.6%

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

   12. Simplified 15.6%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.42 \cdot 10^{-34}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 320:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14: 46.5% 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 91.3%

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

   1. associate--l+ 91.3%

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

   2. +-commutative 91.3%

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

   3. associate-+l- 91.3%

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

   4. *-commutative 91.3%

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

   5. *-commutative 91.3%

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

   6. sub-neg 91.3%

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

   7. metadata-eval 91.3%

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

   8. sub-neg 91.3%

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

   9. log1p-def 99.8%

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

   10. sub-neg 99.8%

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

   11. +-commutative 99.8%

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

   12. *-commutative 99.8%

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

   13. distribute-rgt-neg-in 99.8%

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

   14. fma-def 99.8%

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

   15. neg-sub0 99.8%

       \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{0 - \left(x - 1\right)}, t\right) \]

   16. associate--r- 99.8%

       \[\leadsto \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \mathsf{fma}\left(\log y, \color{blue}{\left(0 - x\right) + 1}, t\right) \]

   17. neg-sub0 99.8%

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

   18. +-commutative 99.8%

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

   19. unsub-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 99.5%

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

   1. associate-*r* 99.5%

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

   2. *-commutative 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. +-commutative 99.5%

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

   6. distribute-lft-in 99.5%

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

   7. metadata-eval 99.5%

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

   8. neg-mul-1 99.5%

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

   9. unsub-neg 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in t around inf 42.9%

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

8. Final simplification 42.9%

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

Alternative 15: 35.8% 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 91.3%

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

2. Taylor expanded in y around 0 90.9%

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

3. Taylor expanded in t around inf 34.1%

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

   1. neg-mul-1 34.1%

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

5. Simplified 34.1%

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

6. Final simplification 34.1%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023192 
(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))