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

Percentage Accurate: 89.3% → 99.8%

Time: 22.4s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

   1. +-commutative 89.4%

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

   2. fma-def 89.4%

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

   3. sub-neg 89.4%

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

   4. metadata-eval 89.4%

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

   5. sub-neg 89.4%

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

   6. log1p-def 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in y around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

   7. distribute-lft-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. neg-mul-1 99.0%

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

   10. unsub-neg 99.0%

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

5. Simplified 99.0%

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

   1. *-commutative 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. distribute-lft-in 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 3: 88.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1 \lor \neg \left(-1 + x \leq -0.9999995\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -1.0) (not (<= (+ -1.0 x) -0.9999995)))
   (- (* (log y) (+ -1.0 x)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.0) || !((-1.0 + x) <= -0.9999995)) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-1.0d0)) .or. (.not. (((-1.0d0) + x) <= (-0.9999995d0)))) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.0) || !((-1.0 + x) <= -0.9999995)) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -1.0) or not ((-1.0 + x) <= -0.9999995):
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -1.0) || !(Float64(-1.0 + x) <= -0.9999995))
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -1.0) || ~(((-1.0 + x) <= -0.9999995)))
		tmp = (log(y) * (-1.0 + x)) - t;
	else
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.9999995]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -1 or -0.999999500000000041 < (-.f64 x 1)

   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. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-- 89.4%

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

      2. log-div 89.4%

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

      3. metadata-eval 89.4%

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

      4. pow3 89.4%

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

      5. sub-neg 89.4%

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

      6. distribute-rgt-neg-out 89.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 89.4%

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

      9. sqr-neg 89.4%

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

      10. sqrt-unprod 89.4%

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

      11. add-sqr-sqrt 89.4%

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

      12. log1p-udef 89.4%

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

      13. pow3 89.4%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      14. metadata-eval 89.4%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      15. log1p-udef 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

      16. *-un-lft-identity 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

      17. fma-def 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right)\right) - t \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   5. Taylor expanded in y around 0 87.7%

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

   if -1 < (-.f64 x 1) < -0.999999500000000041

   1. Initial program 84.7%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. neg-mul-1 100.0%

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

      10. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. log-rec 100.0%

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

      3. +-commutative 100.0%

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

      4. log-rec 100.0%

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

      5. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1 \lor \neg \left(-1 + x \leq -0.9999995\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -200 \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;\left(x \cdot \log y - z \cdot y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -200.0) (not (<= (+ -1.0 x) -0.5)))
   (- (- (* x (log y)) (* z y)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -200.0) || !((-1.0 + x) <= -0.5)) {
		tmp = ((x * log(y)) - (z * y)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-200.0d0)) .or. (.not. (((-1.0d0) + x) <= (-0.5d0)))) then
        tmp = ((x * log(y)) - (z * y)) - t
    else
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -200.0) || !((-1.0 + x) <= -0.5)) {
		tmp = ((x * Math.log(y)) - (z * y)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -200.0) or not ((-1.0 + x) <= -0.5):
		tmp = ((x * math.log(y)) - (z * y)) - t
	else:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -200.0) || !(Float64(-1.0 + x) <= -0.5))
		tmp = Float64(Float64(Float64(x * log(y)) - Float64(z * y)) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -200.0) || ~(((-1.0 + x) <= -0.5)))
		tmp = ((x * log(y)) - (z * y)) - t;
	else
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -200.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.5]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -200 or -0.5 < (-.f64 x 1)

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. distribute-rgt-neg-in 99.1%

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

      3. mul-1-neg 99.1%

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

      4. sub-neg 99.1%

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

      5. metadata-eval 99.1%

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

      6. +-commutative 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. neg-mul-1 99.1%

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

      10. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf 97.4%

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

      1. *-commutative 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in z around inf 97.4%

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

       1. associate-*r* 97.4%

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

       2. neg-mul-1 97.4%

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

   11. Simplified 97.4%

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

   if -200 < (-.f64 x 1) < -0.5

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. distribute-rgt-neg-in 98.8%

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

      3. mul-1-neg 98.8%

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

      4. sub-neg 98.8%

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

      5. metadata-eval 98.8%

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

      6. +-commutative 98.8%

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

      7. distribute-lft-in 98.8%

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

      8. metadata-eval 98.8%

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

      9. neg-mul-1 98.8%

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

      10. unsub-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around 0 97.6%

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

      1. neg-mul-1 97.6%

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

      2. log-rec 97.6%

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

      3. +-commutative 97.6%

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

      4. log-rec 97.6%

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

      5. unsub-neg 97.6%

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

   8. Simplified 97.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -200 \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;\left(x \cdot \log y - z \cdot y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+202} \lor \neg \left(z \leq 1.3 \cdot 10^{+186}\right):\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.9e+202) (not (<= z 1.3e+186)))
   (- (* z (log1p (- y))) t)
   (- (* (log y) (+ -1.0 x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.9e+202) || !(z <= 1.3e+186)) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = (log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.9e+202) || !(z <= 1.3e+186)) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.9e+202) or not (z <= 1.3e+186):
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.9e+202) || !(z <= 1.3e+186))
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.9e+202], N[Not[LessEqual[z, 1.3e+186]], $MachinePrecision]], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.89999999999999983e202 or 1.3e186 < z

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. sub-neg 55.3%

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

      3. metadata-eval 55.3%

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

      4. flip-+ 44.3%

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

      5. associate-*r/ 44.3%

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

      6. metadata-eval 44.3%

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

      7. fma-neg 44.3%

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

      8. metadata-eval 44.3%

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

      9. sub-neg 44.3%

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

      10. metadata-eval 44.3%

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

      11. +-commutative 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. associate-/l* 44.3%

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

   6. Simplified 44.3%

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

   7. Taylor expanded in z around inf 42.9%

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

      1. sub-neg 42.9%

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

      2. log1p-def 88.0%

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

   9. Simplified 88.0%

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

   if -3.89999999999999983e202 < z < 1.3e186

   1. Initial program 96.5%

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

      1. flip3-- 96.3%

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

      2. log-div 96.3%

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

      3. metadata-eval 96.3%

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

      4. pow3 96.3%

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

      5. sub-neg 96.3%

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

      6. distribute-rgt-neg-out 96.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 96.3%

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

      9. sqr-neg 96.3%

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

      10. sqrt-unprod 96.3%

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

      11. add-sqr-sqrt 96.3%

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

      12. log1p-udef 96.3%

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

      13. pow3 96.3%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      14. metadata-eval 96.3%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      15. log1p-udef 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

      16. *-un-lft-identity 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

      17. fma-def 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right)\right) - t \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   5. Taylor expanded in y around 0 95.3%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+202} \lor \neg \left(z \leq 1.3 \cdot 10^{+186}\right):\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.52 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.52) (not (<= x 1.0))) (- (* x (log y)) t) (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.52) || !(x <= 1.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-0.52d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.52) || !(x <= 1.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -0.52) or not (x <= 1.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.52) || !(x <= 1.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -0.52) || ~((x <= 1.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.52], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.52000000000000002 or 1 < x

   1. Initial program 94.2%

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

      1. flip3-- 94.1%

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

      2. log-div 94.1%

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

      3. metadata-eval 94.1%

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

      4. pow3 94.1%

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

      5. sub-neg 94.1%

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

      6. distribute-rgt-neg-out 94.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 94.1%

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

      9. sqr-neg 94.1%

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

      10. sqrt-unprod 94.1%

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

      11. add-sqr-sqrt 94.1%

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

      12. log1p-udef 94.1%

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

      13. pow3 94.1%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      14. metadata-eval 94.1%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      15. log1p-udef 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

      16. *-un-lft-identity 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

      17. fma-def 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right)\right) - t \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   5. Taylor expanded in x around inf 91.7%

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

   if -0.52000000000000002 < x < 1

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

      2. sub-neg 85.1%

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

      3. metadata-eval 85.1%

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

      4. flip-+ 85.1%

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

      5. associate-*r/ 85.1%

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

      6. metadata-eval 85.1%

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

      7. fma-neg 85.1%

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

      8. metadata-eval 85.1%

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

      9. sub-neg 85.1%

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

      10. metadata-eval 85.1%

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

      11. +-commutative 85.1%

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

   4. Applied egg-rr 85.1%

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

      1. associate-/l* 85.1%

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

   6. Simplified 85.1%

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

   7. Taylor expanded in y around 0 82.6%

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

      1. associate-/l* 82.6%

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

      2. unpow2 82.6%

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

      3. fma-neg 82.6%

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

      4. metadata-eval 82.6%

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

   9. Simplified 82.6%

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

   10. Taylor expanded in x around 0 81.4%

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

       1. neg-mul-1 81.4%

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

   12. Simplified 81.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.52 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.52 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.52) (not (<= x 1.0)))
   (- (* x (log y)) t)
   (- (- y (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.52) || !(x <= 1.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-0.52d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.52) || !(x <= 1.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -0.52) or not (x <= 1.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.52) || !(x <= 1.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -0.52) || ~((x <= 1.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.52], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.52000000000000002 or 1 < x

   1. Initial program 94.2%

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

      1. flip3-- 94.1%

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

      2. log-div 94.1%

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

      3. metadata-eval 94.1%

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

      4. pow3 94.1%

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

      5. sub-neg 94.1%

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

      6. distribute-rgt-neg-out 94.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 94.1%

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

      9. sqr-neg 94.1%

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

      10. sqrt-unprod 94.1%

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

      11. add-sqr-sqrt 94.1%

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

      12. log1p-udef 94.1%

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

      13. pow3 94.1%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      14. metadata-eval 94.1%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      15. log1p-udef 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

      16. *-un-lft-identity 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

      17. fma-def 99.7%

          \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right)\right) - t \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   5. Taylor expanded in x around inf 91.7%

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

   if -0.52000000000000002 < x < 1

   1. Initial program 85.1%

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

   3. Taylor expanded in y around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. distribute-rgt-neg-in 98.8%

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

      3. mul-1-neg 98.8%

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

      4. sub-neg 98.8%

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

      5. metadata-eval 98.8%

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

      6. +-commutative 98.8%

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

      7. distribute-lft-in 98.8%

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

      8. metadata-eval 98.8%

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

      9. neg-mul-1 98.8%

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

      10. unsub-neg 98.8%

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

   5. Simplified 98.8%

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

      1. sub-neg 98.8%

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

      2. metadata-eval 98.8%

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

      3. expm1-log1p-u 98.4%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + y \cdot \left(1 - z\right)\right) - t \]

   7. Applied egg-rr 98.4%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + y \cdot \left(1 - z\right)\right) - t \]

   8. Taylor expanded in x around 0 97.1%

      \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\log \left(1 + -1 \cdot \log y\right)}\right) + y \cdot \left(1 - z\right)\right) - t \]
   9. Step-by-step derivation

      1. log1p-def 97.1%

         \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot \log y\right)}\right) + y \cdot \left(1 - z\right)\right) - t \]

      2. neg-mul-1 97.1%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-\log y}\right)\right) + y \cdot \left(1 - z\right)\right) - t \]

   10. Simplified 97.1%

       \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-\log y\right)}\right) + y \cdot \left(1 - z\right)\right) - t \]

   11. Taylor expanded in z around 0 81.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.52 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+171} \lor \neg \left(z \leq 6.5 \cdot 10^{+20}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.25e+171) (not (<= z 6.5e+20)))
   (- (* z (- y)) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e+171) || !(z <= 6.5e+20)) {
		tmp = (z * -y) - t;
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.25d+171)) .or. (.not. (z <= 6.5d+20))) then
        tmp = (z * -y) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e+171) || !(z <= 6.5e+20)) {
		tmp = (z * -y) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.25e+171) or not (z <= 6.5e+20):
		tmp = (z * -y) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.25e+171) || !(z <= 6.5e+20))
		tmp = Float64(Float64(z * Float64(-y)) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.25e+171) || ~((z <= 6.5e+20)))
		tmp = (z * -y) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.25e+171], N[Not[LessEqual[z, 6.5e+20]], $MachinePrecision]], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2500000000000001e171 or 6.5e20 < z

   1. Initial program 68.6%

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

   3. Taylor expanded in y around 0 97.3%

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

      1. mul-1-neg 97.3%

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

      2. distribute-rgt-neg-in 97.3%

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

      3. mul-1-neg 97.3%

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

      4. sub-neg 97.3%

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

      5. metadata-eval 97.3%

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

      6. +-commutative 97.3%

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

      7. distribute-lft-in 97.3%

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

      8. metadata-eval 97.3%

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

      9. neg-mul-1 97.3%

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

      10. unsub-neg 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in x around inf 93.7%

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

      1. *-commutative 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in z around inf 71.8%

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

       1. mul-1-neg 71.8%

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

       2. distribute-rgt-neg-in 71.8%

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

   11. Simplified 71.8%

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

   if -1.2500000000000001e171 < z < 6.5e20

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

      4. flip-+ 75.7%

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

      5. associate-*r/ 75.7%

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

      6. metadata-eval 75.7%

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

      7. fma-neg 75.7%

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

      8. metadata-eval 75.7%

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

      9. sub-neg 75.7%

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

      10. metadata-eval 75.7%

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

      11. +-commutative 75.7%

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

   4. Applied egg-rr 75.7%

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

      1. associate-/l* 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in y around 0 75.2%

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

      1. associate-/l* 75.2%

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

      2. unpow2 75.2%

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

      3. fma-neg 75.2%

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

      4. metadata-eval 75.2%

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

   9. Simplified 75.2%

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

   10. Taylor expanded in x around 0 64.1%

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

       1. neg-mul-1 64.1%

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

   12. Simplified 64.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+171} \lor \neg \left(z \leq 6.5 \cdot 10^{+20}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in y around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

   7. distribute-lft-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. neg-mul-1 99.0%

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

   10. unsub-neg 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 10: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in y around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

   7. distribute-lft-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. neg-mul-1 99.0%

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

   10. unsub-neg 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in z around inf 98.7%

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

   1. associate-*r* 81.9%

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

   2. neg-mul-1 81.9%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 11: 45.5% 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 89.4%

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

   1. *-commutative 89.4%

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

   2. sub-neg 89.4%

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

   3. metadata-eval 89.4%

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

   4. flip-+ 67.2%

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

   5. associate-*r/ 67.2%

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

   6. metadata-eval 67.2%

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

   7. fma-neg 67.2%

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

   8. metadata-eval 67.2%

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

   9. sub-neg 67.2%

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

   10. metadata-eval 67.2%

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

   11. +-commutative 67.2%

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

4. Applied egg-rr 67.2%

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

   1. associate-/l* 67.2%

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

6. Simplified 67.2%

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

   1. div-inv 67.2%

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

   2. fma-def 67.2%

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

   3. clear-num 67.2%

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

   4. metadata-eval 67.2%

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

   5. fma-neg 67.2%

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

   6. metadata-eval 67.2%

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

   7. +-commutative 67.2%

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

   8. flip-- 89.4%

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

   9. sub-neg 89.4%

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

   10. metadata-eval 89.4%

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

   11. sub-neg 89.4%

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

   12. metadata-eval 89.4%

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

   13. sub-neg 89.4%

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

   14. log1p-def 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in y around 0 99.0%

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

    1. mul-1-neg 99.0%

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

    2. distribute-rgt-neg-in 99.0%

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

    3. sub-neg 99.0%

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

    4. metadata-eval 99.0%

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

    5. +-commutative 99.0%

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

    6. distribute-neg-in 99.0%

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

    7. metadata-eval 99.0%

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

    8. sub-neg 99.0%

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

11. Simplified 99.0%

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

12. Taylor expanded in y around inf 52.6%

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

13. Final simplification 52.6%

    \[\leadsto y \cdot \left(1 - z\right) - t \]
14. Add Preprocessing

Alternative 12: 45.3% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in y around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

   7. distribute-lft-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. neg-mul-1 99.0%

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

   10. unsub-neg 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around inf 82.0%

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

   1. *-commutative 82.0%

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

8. Simplified 82.0%

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

9. Taylor expanded in z around inf 52.4%

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

    1. mul-1-neg 52.4%

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

    2. distribute-rgt-neg-in 52.4%

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

11. Simplified 52.4%

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

12. Final simplification 52.4%

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

Alternative 13: 35.1% 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 89.4%

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

   1. flip3-- 89.3%

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

   2. log-div 89.2%

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

   3. metadata-eval 89.2%

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

   4. pow3 89.2%

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

   5. sub-neg 89.2%

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

   6. distribute-rgt-neg-out 89.2%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 89.3%

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

   9. sqr-neg 89.3%

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

   10. sqrt-unprod 89.3%

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

   11. add-sqr-sqrt 89.3%

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

   12. log1p-udef 89.2%

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

   13. pow3 89.2%

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   14. metadata-eval 89.2%

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   15. log1p-udef 99.7%

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

   16. *-un-lft-identity 99.7%

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

   17. fma-def 99.7%

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right)\right) - t \]

4. Applied egg-rr 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

5. Taylor expanded in x around inf 71.3%

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

6. Taylor expanded in x around 0 41.7%

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

   1. neg-mul-1 41.7%

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

8. Simplified 41.7%

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

9. Final simplification 41.7%

   \[\leadsto -t \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))