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

Percentage Accurate: 89.0% → 99.8%

Time: 21.6s

Alternatives: 18

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. +-commutative 89.2%

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

   2. fma-define 89.2%

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

   3. sub-neg 89.2%

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

   4. metadata-eval 89.2%

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

   5. sub-neg 89.2%

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

   6. log1p-define 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%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in x around 0 89.2%

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

   1. associate-+r+ 89.2%

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

   2. distribute-rgt-out 89.2%

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

   3. +-commutative 89.2%

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

   4. fma-define 89.2%

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

   5. +-commutative 89.2%

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

   6. sub-neg 89.2%

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

   7. mul-1-neg 89.2%

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

   8. log1p-define 99.9%

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

   9. mul-1-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. metadata-eval 99.9%

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

   12. +-commutative 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in x around 0 89.2%

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

   1. associate-+r+ 89.2%

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

   2. distribute-rgt-out 89.2%

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

   3. +-commutative 89.2%

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

   4. fma-define 89.2%

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

   5. +-commutative 89.2%

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

   6. sub-neg 89.2%

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

   7. mul-1-neg 89.2%

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

   8. log1p-define 99.9%

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

   9. mul-1-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. metadata-eval 99.9%

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

   12. +-commutative 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in y around 0 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

   5. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in y around 0 99.6%

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

    1. fma-define 99.6%

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

    2. sub-neg 99.6%

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

    3. metadata-eval 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000005 \lor \neg \left(-1 + x \leq 5000000000\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -1.00000005) (not (<= (+ -1.0 x) 5000000000.0)))
   (- (* (+ -1.0 x) (log y)) t)
   (- (- (- y (* z y)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.00000005) || !((-1.0 + x) <= 5000000000.0)) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = ((y - (z * y)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-1.00000005d0)) .or. (.not. (((-1.0d0) + x) <= 5000000000.0d0))) then
        tmp = (((-1.0d0) + x) * log(y)) - t
    else
        tmp = ((y - (z * y)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.00000005) || !((-1.0 + x) <= 5000000000.0)) {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	} else {
		tmp = ((y - (z * y)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -1.00000005) or not ((-1.0 + x) <= 5000000000.0):
		tmp = ((-1.0 + x) * math.log(y)) - t
	else:
		tmp = ((y - (z * y)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -1.00000005) || !(Float64(-1.0 + x) <= 5000000000.0))
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(y - Float64(z * y)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -1.00000005) || ~(((-1.0 + x) <= 5000000000.0)))
		tmp = ((-1.0 + x) * log(y)) - t;
	else
		tmp = ((y - (z * y)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.00000005], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], 5000000000.0]], $MachinePrecision]], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -1.00000004999999992 or 5e9 < (-.f64 x 1)

   1. Initial program 92.0%

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

      1. +-commutative 92.0%

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

      2. fma-define 92.0%

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

      3. sub-neg 92.0%

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

      4. metadata-eval 92.0%

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

      5. sub-neg 92.0%

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

      6. log1p-define 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 91.8%

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

   if -1.00000004999999992 < (-.f64 x 1) < 5e9

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. sub-neg 85.6%

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

      3. metadata-eval 85.6%

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

      4. fma-define 85.6%

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

      5. sub-neg 85.6%

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

      6. mul-1-neg 85.6%

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

      7. log1p-define 98.7%

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

      8. mul-1-neg 98.7%

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

      9. +-commutative 98.7%

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

      10. mul-1-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in y around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. log-rec 98.2%

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

      3. +-commutative 98.2%

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

      4. log-rec 98.2%

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

      5. unsub-neg 98.2%

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

      6. mul-1-neg 98.2%

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

      7. distribute-rgt-neg-in 98.2%

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

      8. sub-neg 98.2%

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

      9. metadata-eval 98.2%

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

      10. +-commutative 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in z around 0 98.2%

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

       1. mul-1-neg 98.2%

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

       2. unsub-neg 98.2%

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

   11. Simplified 98.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000005 \lor \neg \left(-1 + x \leq 5000000000\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-1 + x\right) \cdot \log y\\ \mathbf{if}\;t \leq -9 \cdot 10^{+15} \lor \neg \left(t \leq 2 \cdot 10^{-21}\right):\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\_1\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ -1.0 x) (log y))))
   (if (or (<= t -9e+15) (not (<= t 2e-21))) (- t_1 t) (- (+ y t_1) (* z y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-1.0 + x) * log(y);
	double tmp;
	if ((t <= -9e+15) || !(t <= 2e-21)) {
		tmp = t_1 - t;
	} else {
		tmp = (y + t_1) - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-1.0d0) + x) * log(y)
    if ((t <= (-9d+15)) .or. (.not. (t <= 2d-21))) then
        tmp = t_1 - t
    else
        tmp = (y + t_1) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-1.0 + x) * Math.log(y);
	double tmp;
	if ((t <= -9e+15) || !(t <= 2e-21)) {
		tmp = t_1 - t;
	} else {
		tmp = (y + t_1) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-1.0 + x) * math.log(y)
	tmp = 0
	if (t <= -9e+15) or not (t <= 2e-21):
		tmp = t_1 - t
	else:
		tmp = (y + t_1) - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-1.0 + x) * log(y))
	tmp = 0.0
	if ((t <= -9e+15) || !(t <= 2e-21))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(Float64(y + t_1) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-1.0 + x) * log(y);
	tmp = 0.0;
	if ((t <= -9e+15) || ~((t <= 2e-21)))
		tmp = t_1 - t;
	else
		tmp = (y + t_1) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -9e+15], N[Not[LessEqual[t, 2e-21]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(N[(y + t$95$1), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -9e15 or 1.99999999999999982e-21 < t

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

      2. fma-define 97.7%

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

      3. sub-neg 97.7%

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

      4. metadata-eval 97.7%

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

      5. sub-neg 97.7%

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

      6. log1p-define 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. Taylor expanded in y around 0 97.6%

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

   if -9e15 < t < 1.99999999999999982e-21

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0 81.3%

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

      1. associate-+r+ 81.3%

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

      2. distribute-rgt-out 81.3%

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

      3. +-commutative 81.3%

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

      4. fma-define 81.3%

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

      5. +-commutative 81.3%

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

      6. sub-neg 81.3%

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

      7. mul-1-neg 81.3%

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

      8. log1p-define 99.8%

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

      9. mul-1-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-rgt-neg-in 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

      5. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in z around 0 99.3%

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

       1. mul-1-neg 55.6%

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

       2. unsub-neg 55.6%

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

   11. Simplified 99.3%

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

   12. Taylor expanded in t around 0 98.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+15} \lor \neg \left(t \leq 2 \cdot 10^{-21}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(-1 + x\right) \cdot \log y\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-1 + x\right) \cdot \log y\\ \mathbf{if}\;t \leq -9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_1}} - t\\ \mathbf{elif}\;t \leq 10^{-21}:\\ \;\;\;\;\left(y + t\_1\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ -1.0 x) (log y))))
   (if (<= t -9e+15)
     (- (/ 1.0 (/ 1.0 t_1)) t)
     (if (<= t 1e-21) (- (+ y t_1) (* z y)) (- t_1 t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-1.0 + x) * log(y);
	double tmp;
	if (t <= -9e+15) {
		tmp = (1.0 / (1.0 / t_1)) - t;
	} else if (t <= 1e-21) {
		tmp = (y + t_1) - (z * y);
	} else {
		tmp = t_1 - 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) :: t_1
    real(8) :: tmp
    t_1 = ((-1.0d0) + x) * log(y)
    if (t <= (-9d+15)) then
        tmp = (1.0d0 / (1.0d0 / t_1)) - t
    else if (t <= 1d-21) then
        tmp = (y + t_1) - (z * y)
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-1.0 + x) * Math.log(y);
	double tmp;
	if (t <= -9e+15) {
		tmp = (1.0 / (1.0 / t_1)) - t;
	} else if (t <= 1e-21) {
		tmp = (y + t_1) - (z * y);
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-1.0 + x) * math.log(y)
	tmp = 0
	if t <= -9e+15:
		tmp = (1.0 / (1.0 / t_1)) - t
	elif t <= 1e-21:
		tmp = (y + t_1) - (z * y)
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-1.0 + x) * log(y))
	tmp = 0.0
	if (t <= -9e+15)
		tmp = Float64(Float64(1.0 / Float64(1.0 / t_1)) - t);
	elseif (t <= 1e-21)
		tmp = Float64(Float64(y + t_1) - Float64(z * y));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-1.0 + x) * log(y);
	tmp = 0.0;
	if (t <= -9e+15)
		tmp = (1.0 / (1.0 / t_1)) - t;
	elseif (t <= 1e-21)
		tmp = (y + t_1) - (z * y);
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9e+15], N[(N[(1.0 / N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 1e-21], N[(N[(y + t$95$1), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9e15

   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. Step-by-step derivation

      1. +-commutative 99.1%

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

      2. fma-define 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. sub-neg 99.1%

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

      6. log1p-define 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. Taylor expanded in y around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. +-commutative 99.1%

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

      6. metadata-eval 99.1%

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

      7. sub-neg 99.1%

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

      8. flip-- 72.9%

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

      9. metadata-eval 72.9%

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

      10. fma-neg 72.9%

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

      11. metadata-eval 72.9%

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

      12. +-commutative 72.9%

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

      13. associate-/r/ 72.9%

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

      14. clear-num 72.9%

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

      15. clear-num 72.9%

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

      16. associate-/r/ 72.9%

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

      17. metadata-eval 72.9%

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

      18. fma-neg 72.9%

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

      19. metadata-eval 72.9%

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

      20. +-commutative 72.9%

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

      21. flip-- 99.1%

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

      22. sub-neg 99.1%

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

      23. metadata-eval 99.1%

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

      24. +-commutative 99.1%

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

      25. *-commutative 99.1%

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

   7. Applied egg-rr 99.1%

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

   if -9e15 < t < 9.99999999999999908e-22

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0 81.3%

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

      1. associate-+r+ 81.3%

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

      2. distribute-rgt-out 81.3%

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

      3. +-commutative 81.3%

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

      4. fma-define 81.3%

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

      5. +-commutative 81.3%

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

      6. sub-neg 81.3%

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

      7. mul-1-neg 81.3%

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

      8. log1p-define 99.8%

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

      9. mul-1-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-rgt-neg-in 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

      5. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in z around 0 99.3%

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

       1. mul-1-neg 55.6%

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

       2. unsub-neg 55.6%

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

   11. Simplified 99.3%

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

   12. Taylor expanded in t around 0 98.5%

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

   if 9.99999999999999908e-22 < t

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. fma-define 96.6%

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

      3. sub-neg 96.6%

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

      4. metadata-eval 96.6%

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

      5. sub-neg 96.6%

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

      6. log1p-define 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. Taylor expanded in y around 0 96.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(-1 + x\right) \cdot \log y}} - t\\ \mathbf{elif}\;t \leq 10^{-21}:\\ \;\;\;\;\left(y + \left(-1 + x\right) \cdot \log y\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 4.35 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 4.35e-13)))
   (- (* x (log y)) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 4.35e-13)) {
		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 <= (-1.0d0)) .or. (.not. (x <= 4.35d-13))) 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 <= -1.0) || !(x <= 4.35e-13)) {
		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 <= -1.0) or not (x <= 4.35e-13):
		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 <= -1.0) || !(x <= 4.35e-13))
		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 <= -1.0) || ~((x <= 4.35e-13)))
		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, -1.0], N[Not[LessEqual[x, 4.35e-13]], $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 < -1 or 4.35000000000000014e-13 < x

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. associate-+r+ 90.8%

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

      2. distribute-rgt-out 90.9%

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

      3. +-commutative 90.9%

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

      4. fma-define 90.9%

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

      5. +-commutative 90.9%

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

      6. sub-neg 90.9%

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

      7. mul-1-neg 90.9%

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

      8. log1p-define 99.8%

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

      9. mul-1-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 89.7%

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

      1. *-commutative 89.7%

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

   8. Simplified 89.7%

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

   if -1 < x < 4.35000000000000014e-13

   1. Initial program 87.5%

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

      1. +-commutative 87.5%

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

      2. fma-define 87.5%

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

      3. sub-neg 87.5%

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

      4. metadata-eval 87.5%

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

      5. sub-neg 87.5%

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

      6. log1p-define 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 86.7%

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

   6. Taylor expanded in x around 0 86.4%

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

      1. mul-1-neg 86.4%

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

   8. Simplified 86.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 4.35 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 4.35 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 4.35e-13)))
   (- (* x (log y)) t)
   (- (- y (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 4.35e-13)) {
		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 <= (-1.0d0)) .or. (.not. (x <= 4.35d-13))) 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 <= -1.0) || !(x <= 4.35e-13)) {
		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 <= -1.0) or not (x <= 4.35e-13):
		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 <= -1.0) || !(x <= 4.35e-13))
		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 <= -1.0) || ~((x <= 4.35e-13)))
		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, -1.0], N[Not[LessEqual[x, 4.35e-13]], $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 < -1 or 4.35000000000000014e-13 < x

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. associate-+r+ 90.8%

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

      2. distribute-rgt-out 90.9%

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

      3. +-commutative 90.9%

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

      4. fma-define 90.9%

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

      5. +-commutative 90.9%

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

      6. sub-neg 90.9%

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

      7. mul-1-neg 90.9%

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

      8. log1p-define 99.8%

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

      9. mul-1-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 89.7%

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

      1. *-commutative 89.7%

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

   8. Simplified 89.7%

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

   if -1 < x < 4.35000000000000014e-13

   1. Initial program 87.5%

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

   3. Taylor expanded in x around 0 87.1%

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

      1. +-commutative 87.1%

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

      2. sub-neg 87.1%

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

      3. metadata-eval 87.1%

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

      4. fma-define 87.1%

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

      5. sub-neg 87.1%

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

      6. mul-1-neg 87.1%

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

      7. log1p-define 99.6%

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

      8. mul-1-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. mul-1-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. log-rec 99.1%

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

      3. +-commutative 99.1%

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

      4. log-rec 99.1%

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

      5. unsub-neg 99.1%

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

      6. mul-1-neg 99.1%

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

      7. distribute-rgt-neg-in 99.1%

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

      8. sub-neg 99.1%

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

      9. metadata-eval 99.1%

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

      10. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in z around 0 86.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 4.35 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+129} \lor \neg \left(z \leq 6.2 \cdot 10^{+178}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4e+129) (not (<= z 6.2e+178)))
   (- (- t) (* z y))
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4e+129) || !(z <= 6.2e+178)) {
		tmp = -t - (z * y);
	} 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 <= (-4d+129)) .or. (.not. (z <= 6.2d+178))) then
        tmp = -t - (z * y)
    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 <= -4e+129) || !(z <= 6.2e+178)) {
		tmp = -t - (z * y);
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4e+129) or not (z <= 6.2e+178):
		tmp = -t - (z * y)
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4e+129) || !(z <= 6.2e+178))
		tmp = Float64(Float64(-t) - Float64(z * y));
	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 <= -4e+129) || ~((z <= 6.2e+178)))
		tmp = -t - (z * y);
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4e+129], N[Not[LessEqual[z, 6.2e+178]], $MachinePrecision]], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4e129 or 6.19999999999999982e178 < z

   1. Initial program 63.0%

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

   3. Taylor expanded in x around 0 63.0%

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

      1. associate-+r+ 63.0%

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

      2. distribute-rgt-out 63.0%

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

      3. +-commutative 63.0%

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

      4. fma-define 63.0%

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

      5. +-commutative 63.0%

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

      6. sub-neg 63.0%

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

      7. mul-1-neg 63.0%

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

      8. log1p-define 99.9%

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

      9. mul-1-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

      5. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in z around inf 63.4%

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

       1. mul-1-neg 63.4%

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

       2. distribute-rgt-neg-in 63.4%

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

   11. Simplified 63.4%

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

   if -4e129 < z < 6.19999999999999982e178

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

      2. fma-define 98.5%

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

      3. sub-neg 98.5%

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

      4. metadata-eval 98.5%

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

      5. sub-neg 98.5%

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

      6. log1p-define 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. Taylor expanded in y around 0 98.0%

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

   6. Taylor expanded in x around 0 64.7%

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

      1. mul-1-neg 64.7%

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

   8. Simplified 64.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+129} \lor \neg \left(z \leq 6.2 \cdot 10^{+178}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

   4. *-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. unsub-neg 99.6%

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

   7. *-commutative 99.6%

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

   8. +-commutative 99.6%

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

   9. sub-neg 99.6%

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

   10. metadata-eval 99.6%

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

   11. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 11: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-19} \lor \neg \left(t \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -7.6e-19) (not (<= t 3.2e-7))) (- (- t) (* z y)) (- (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.6e-19) || !(t <= 3.2e-7)) {
		tmp = -t - (z * y);
	} else {
		tmp = -log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-7.6d-19)) .or. (.not. (t <= 3.2d-7))) then
        tmp = -t - (z * y)
    else
        tmp = -log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.6e-19) || !(t <= 3.2e-7)) {
		tmp = -t - (z * y);
	} else {
		tmp = -Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -7.6e-19) or not (t <= 3.2e-7):
		tmp = -t - (z * y)
	else:
		tmp = -math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -7.6e-19) || !(t <= 3.2e-7))
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(-log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -7.6e-19) || ~((t <= 3.2e-7)))
		tmp = -t - (z * y);
	else
		tmp = -log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -7.6e-19], N[Not[LessEqual[t, 3.2e-7]], $MachinePrecision]], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], (-N[Log[y], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if t < -7.6e-19 or 3.2000000000000001e-7 < t

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0 94.8%

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

      1. associate-+r+ 94.8%

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

      2. distribute-rgt-out 94.8%

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

      3. +-commutative 94.8%

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

      4. fma-define 94.8%

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

      5. +-commutative 94.8%

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

      6. sub-neg 94.8%

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

      7. mul-1-neg 94.8%

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

      8. log1p-define 99.9%

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

      9. mul-1-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in z around inf 77.1%

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

       1. mul-1-neg 77.1%

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

       2. distribute-rgt-neg-in 77.1%

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

   11. Simplified 77.1%

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

   if -7.6e-19 < t < 3.2000000000000001e-7

   1. Initial program 83.5%

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

      1. +-commutative 83.5%

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

      2. fma-define 83.5%

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

      3. sub-neg 83.5%

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

      4. metadata-eval 83.5%

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

      5. sub-neg 83.5%

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

      6. log1p-define 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. Taylor expanded in y around 0 82.8%

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

   6. Taylor expanded in x around 0 38.3%

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

      1. mul-1-neg 38.3%

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

   8. Simplified 38.3%

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

   9. Taylor expanded in t around 0 38.3%

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

       1. mul-1-neg 38.3%

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

   11. Simplified 38.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-19} \lor \neg \left(t \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+218}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.5e+218) (- (- t) (* z y)) (- (* (+ -1.0 x) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.5e+218) {
		tmp = -t - (z * y);
	} else {
		tmp = ((-1.0 + x) * 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 <= (-1.5d+218)) then
        tmp = -t - (z * y)
    else
        tmp = (((-1.0d0) + x) * 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 <= -1.5e+218) {
		tmp = -t - (z * y);
	} else {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.5e+218:
		tmp = -t - (z * y)
	else:
		tmp = ((-1.0 + x) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.5e+218)
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.5e+218)
		tmp = -t - (z * y);
	else
		tmp = ((-1.0 + x) * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.5e+218], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e218

   1. Initial program 51.9%

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

   3. Taylor expanded in x around 0 51.9%

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

      1. associate-+r+ 51.9%

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

      2. distribute-rgt-out 51.9%

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

      3. +-commutative 51.9%

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

      4. fma-define 51.9%

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

      5. +-commutative 51.9%

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

      6. sub-neg 51.9%

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

      7. mul-1-neg 51.9%

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

      8. log1p-define 100.0%

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

      9. mul-1-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around inf 79.3%

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

       1. mul-1-neg 79.3%

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

       2. distribute-rgt-neg-in 79.3%

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

   11. Simplified 79.3%

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

   if -1.5e218 < z

   1. Initial program 91.7%

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

      1. +-commutative 91.7%

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

      2. fma-define 91.7%

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

      3. sub-neg 91.7%

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

      4. metadata-eval 91.7%

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

      5. sub-neg 91.7%

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

      6. log1p-define 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. Taylor expanded in y around 0 91.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+218}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+15} \lor \neg \left(t \leq 0.0014\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -9e+15) (not (<= t 0.0014))) (- t) (* y (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -9e+15) || !(t <= 0.0014)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-9d+15)) .or. (.not. (t <= 0.0014d0))) then
        tmp = -t
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -9e+15) || !(t <= 0.0014)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -9e+15) or not (t <= 0.0014):
		tmp = -t
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -9e+15) || !(t <= 0.0014))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -9e+15) || ~((t <= 0.0014)))
		tmp = -t;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -9e+15], N[Not[LessEqual[t, 0.0014]], $MachinePrecision]], (-t), N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9e15 or 0.00139999999999999999 < t

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

      2. fma-define 97.7%

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

      3. sub-neg 97.7%

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

      4. metadata-eval 97.7%

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

      5. sub-neg 97.7%

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

      6. log1p-define 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. Taylor expanded in t around inf 76.7%

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

      1. mul-1-neg 76.7%

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

   7. Simplified 76.7%

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

   if -9e15 < t < 0.00139999999999999999

   1. Initial program 81.6%

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

   3. Taylor expanded in x around 0 81.6%

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

      1. associate-+r+ 81.6%

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

      2. distribute-rgt-out 81.6%

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

      3. +-commutative 81.6%

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

      4. fma-define 81.6%

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

      5. +-commutative 81.6%

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

      6. sub-neg 81.6%

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

      7. mul-1-neg 81.6%

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

      8. log1p-define 99.8%

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

      9. mul-1-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in y around 0 99.4%

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

       1. fma-define 99.4%

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

       2. sub-neg 99.4%

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

       3. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in y around inf 20.1%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+15} \lor \neg \left(t \leq 0.0014\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 0.0014:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.25e+16) (- (* z y) t) (if (<= t 0.0014) (* y (- 1.0 z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.25e+16) {
		tmp = (z * y) - t;
	} else if (t <= 0.0014) {
		tmp = y * (1.0 - z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.25d+16)) then
        tmp = (z * y) - t
    else if (t <= 0.0014d0) then
        tmp = y * (1.0d0 - z)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.25e+16) {
		tmp = (z * y) - t;
	} else if (t <= 0.0014) {
		tmp = y * (1.0 - z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.25e+16:
		tmp = (z * y) - t
	elif t <= 0.0014:
		tmp = y * (1.0 - z)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.25e+16)
		tmp = Float64(Float64(z * y) - t);
	elseif (t <= 0.0014)
		tmp = Float64(y * Float64(1.0 - z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.25e+16)
		tmp = (z * y) - t;
	elseif (t <= 0.0014)
		tmp = y * (1.0 - z);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.25e+16], N[(N[(z * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 0.0014], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25e16

   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. Taylor expanded in x around 0 99.1%

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

      1. associate-+r+ 99.1%

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

      2. distribute-rgt-out 99.1%

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

      3. +-commutative 99.1%

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

      4. fma-define 99.1%

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

      5. +-commutative 99.1%

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

      6. sub-neg 99.1%

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

      7. mul-1-neg 99.1%

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

      8. log1p-define 99.9%

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

      9. mul-1-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in z around inf 79.3%

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

       1. mul-1-neg 79.3%

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

       2. distribute-rgt-neg-in 79.3%

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

   11. Simplified 79.3%

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

       1. sub-neg 79.3%

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

       2. add-sqr-sqrt 41.1%

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

       3. sqrt-unprod 61.2%

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

       4. sqr-neg 61.2%

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

       5. sqrt-unprod 38.4%

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

       6. add-sqr-sqrt 79.4%

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

   13. Applied egg-rr 79.4%

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

       1. sub-neg 79.4%

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

   15. Simplified 79.4%

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

   if -1.25e16 < t < 0.00139999999999999999

   1. Initial program 81.6%

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

   3. Taylor expanded in x around 0 81.6%

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

      1. associate-+r+ 81.6%

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

      2. distribute-rgt-out 81.6%

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

      3. +-commutative 81.6%

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

      4. fma-define 81.6%

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

      5. +-commutative 81.6%

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

      6. sub-neg 81.6%

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

      7. mul-1-neg 81.6%

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

      8. log1p-define 99.8%

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

      9. mul-1-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in y around 0 99.4%

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

       1. fma-define 99.4%

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

       2. sub-neg 99.4%

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

       3. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in y around inf 20.1%

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

   if 0.00139999999999999999 < t

   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. Step-by-step derivation

      1. +-commutative 96.5%

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

      2. fma-define 96.5%

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

      3. sub-neg 96.5%

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

      4. metadata-eval 96.5%

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

      5. sub-neg 96.5%

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

      6. log1p-define 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. Taylor expanded in t around inf 74.3%

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

      1. mul-1-neg 74.3%

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

   7. Simplified 74.3%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 0.0014:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+15} \lor \neg \left(t \leq 0.00106\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -9e+15) (not (<= t 0.00106))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -9e+15) || !(t <= 0.00106)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-9d+15)) .or. (.not. (t <= 0.00106d0))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -9e+15) || !(t <= 0.00106)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -9e+15) or not (t <= 0.00106):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -9e+15) || !(t <= 0.00106))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -9e+15) || ~((t <= 0.00106)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -9e+15], N[Not[LessEqual[t, 0.00106]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9e15 or 0.00105999999999999996 < t

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

      2. fma-define 97.7%

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

      3. sub-neg 97.7%

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

      4. metadata-eval 97.7%

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

      5. sub-neg 97.7%

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

      6. log1p-define 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. Taylor expanded in t around inf 76.7%

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

      1. mul-1-neg 76.7%

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

   7. Simplified 76.7%

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

   if -9e15 < t < 0.00105999999999999996

   1. Initial program 81.6%

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

   3. Taylor expanded in x around 0 81.6%

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

      1. associate-+r+ 81.6%

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

      2. distribute-rgt-out 81.6%

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

      3. +-commutative 81.6%

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

      4. fma-define 81.6%

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

      5. +-commutative 81.6%

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

      6. sub-neg 81.6%

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

      7. mul-1-neg 81.6%

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

      8. log1p-define 99.8%

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

      9. mul-1-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in y around 0 99.4%

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

       1. fma-define 99.4%

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

       2. sub-neg 99.4%

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

       3. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in z around inf 19.6%

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

       1. associate-*r* 19.6%

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

       2. neg-mul-1 19.6%

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

   14. Simplified 19.6%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+15} \lor \neg \left(t \leq 0.00106\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.4% 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 89.2%

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

3. Taylor expanded in x around 0 89.2%

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

   1. associate-+r+ 89.2%

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

   2. distribute-rgt-out 89.2%

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

   3. +-commutative 89.2%

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

   4. fma-define 89.2%

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

   5. +-commutative 89.2%

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

   6. sub-neg 89.2%

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

   7. mul-1-neg 89.2%

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

   8. log1p-define 99.9%

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

   9. mul-1-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. metadata-eval 99.9%

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

   12. +-commutative 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in y around 0 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

   5. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in y around inf 47.9%

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

10. Final simplification 47.9%

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

Alternative 17: 47.2% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in x around 0 89.2%

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

   1. associate-+r+ 89.2%

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

   2. distribute-rgt-out 89.2%

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

   3. +-commutative 89.2%

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

   4. fma-define 89.2%

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

   5. +-commutative 89.2%

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

   6. sub-neg 89.2%

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

   7. mul-1-neg 89.2%

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

   8. log1p-define 99.9%

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

   9. mul-1-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. metadata-eval 99.9%

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

   12. +-commutative 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in y around 0 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

   5. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in z around inf 47.7%

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

    1. mul-1-neg 47.7%

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

    2. distribute-rgt-neg-in 47.7%

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

11. Simplified 47.7%

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

12. Final simplification 47.7%

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

Alternative 18: 36.8% accurate, 107.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. +-commutative 89.2%

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

   2. fma-define 89.2%

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

   3. sub-neg 89.2%

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

   4. metadata-eval 89.2%

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

   5. sub-neg 89.2%

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

   6. log1p-define 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. Taylor expanded in t around inf 37.9%

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

   1. mul-1-neg 37.9%

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

7. Simplified 37.9%

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

8. Final simplification 37.9%

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

Reproduce

herbie shell --seed 2024034 
(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))