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

Percentage Accurate: 89.4% → 99.8%

Time: 14.5s

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: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate--l+ 90.6%

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

   2. +-commutative 90.6%

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

   3. associate-+l- 90.6%

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

   4. *-commutative 90.6%

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

   5. *-commutative 90.6%

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

   6. sub-neg 90.6%

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

   7. metadata-eval 90.6%

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

   8. sub-neg 90.6%

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

   9. log1p-def 99.9%

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

   10. sub-neg 99.9%

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

   11. +-commutative 99.9%

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

   12. *-commutative 99.9%

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

   13. distribute-rgt-neg-in 99.9%

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

   14. fma-def 99.9%

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

   15. neg-sub0 99.9%

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

   16. associate--r- 99.9%

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

   17. neg-sub0 99.9%

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

   18. +-commutative 99.9%

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

   19. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(y \cdot \left(y \cdot -0.5\right) - y\right) \cdot \left(z + -1\right) + \log y \cdot \left(-1 + x\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- (* y (* y -0.5)) y) (+ z -1.0)) (* (log y) (+ -1.0 x))) t))

C

double code(double x, double y, double z, double t) {
	return ((((y * (y * -0.5)) - y) * (z + -1.0)) + (log(y) * (-1.0 + x))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((((y * (y * (-0.5d0))) - y) * (z + (-1.0d0))) + (log(y) * ((-1.0d0) + x))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((((y * (y * -0.5)) - y) * (z + -1.0)) + (Math.log(y) * (-1.0 + x))) - t;
}

Python

def code(x, y, z, t):
	return ((((y * (y * -0.5)) - y) * (z + -1.0)) + (math.log(y) * (-1.0 + x))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(Float64(y * Float64(y * -0.5)) - y) * Float64(z + -1.0)) + Float64(log(y) * Float64(-1.0 + x))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((((y * (y * -0.5)) - y) * (z + -1.0)) + (log(y) * (-1.0 + x))) - t;
end

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in y around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. unsub-neg 99.8%

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

   3. *-commutative 99.8%

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

   4. unpow2 99.8%

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

   5. associate-*l* 99.8%

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

4. Simplified 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \left(\left(y \cdot \left(y \cdot -0.5\right) - y\right) \cdot \left(z + -1\right) + \log y \cdot \left(-1 + x\right)\right) - t \]

Alternative 3: 94.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -2 \cdot 10^{+63} \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -2e+63) (not (<= (+ -1.0 x) -0.5)))
   (- (* (log y) (+ -1.0 x)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -2e+63) || !((-1.0 + x) <= -0.5)) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-2d+63)) .or. (.not. (((-1.0d0) + x) <= (-0.5d0)))) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -2e+63) || !((-1.0 + x) <= -0.5)) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -2e+63) or not ((-1.0 + x) <= -0.5):
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -2e+63) || !(Float64(-1.0 + x) <= -0.5))
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -2e+63) || ~(((-1.0 + x) <= -0.5)))
		tmp = (log(y) * (-1.0 + x)) - t;
	else
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -2e+63], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.5]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -2.00000000000000012e63 or -0.5 < (-.f64 x 1)

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 94.1%

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

   if -2.00000000000000012e63 < (-.f64 x 1) < -0.5

   1. Initial program 86.7%

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

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-neg-in 99.5%

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

      8. metadata-eval 99.5%

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

      9. unsub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   8. Simplified 96.5%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -2 \cdot 10^{+63} \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]

Alternative 4: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) + \log y \cdot \left(-1 + x\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* z (- (* -0.5 (* y y)) y)) (* (log y) (+ -1.0 x))) t))

C

double code(double x, double y, double z, double t) {
	return ((z * ((-0.5 * (y * y)) - y)) + (log(y) * (-1.0 + x))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((z * (((-0.5d0) * (y * y)) - y)) + (log(y) * ((-1.0d0) + x))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((z * ((-0.5 * (y * y)) - y)) + (Math.log(y) * (-1.0 + x))) - t;
}

Python

def code(x, y, z, t):
	return ((z * ((-0.5 * (y * y)) - y)) + (math.log(y) * (-1.0 + x))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(z * Float64(Float64(-0.5 * Float64(y * y)) - y)) + Float64(log(y) * Float64(-1.0 + x))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((z * ((-0.5 * (y * y)) - y)) + (log(y) * (-1.0 + x))) - t;
end

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in y around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. unsub-neg 99.8%

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

   3. *-commutative 99.8%

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

   4. unpow2 99.8%

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

   5. associate-*l* 99.8%

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

4. Simplified 99.8%

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

5. Taylor expanded in z around inf 99.8%

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

   1. unpow2 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

   \[\leadsto \left(z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) + \log y \cdot \left(-1 + x\right)\right) - t \]

Alternative 5: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t\\ t_3 := -\log y\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-228}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-99}:\\ \;\;\;\;\left(y - z \cdot y\right) - t\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-81}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (- (- (* (* y y) (+ 0.5 (* z -0.5))) (* y (+ z -1.0))) t))
        (t_3 (- (log y))))
   (if (<= x -2.95e+63)
     t_1
     (if (<= x 2e-246)
       t_2
       (if (<= x 2.1e-228)
         t_3
         (if (<= x 2.15e-99)
           (- (- y (* z y)) t)
           (if (<= x 4.5e-81) t_3 (if (<= x 7.2e+119) t_2 t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	double t_3 = -log(y);
	double tmp;
	if (x <= -2.95e+63) {
		tmp = t_1;
	} else if (x <= 2e-246) {
		tmp = t_2;
	} else if (x <= 2.1e-228) {
		tmp = t_3;
	} else if (x <= 2.15e-99) {
		tmp = (y - (z * y)) - t;
	} else if (x <= 4.5e-81) {
		tmp = t_3;
	} else if (x <= 7.2e+119) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = (((y * y) * (0.5d0 + (z * (-0.5d0)))) - (y * (z + (-1.0d0)))) - t
    t_3 = -log(y)
    if (x <= (-2.95d+63)) then
        tmp = t_1
    else if (x <= 2d-246) then
        tmp = t_2
    else if (x <= 2.1d-228) then
        tmp = t_3
    else if (x <= 2.15d-99) then
        tmp = (y - (z * y)) - t
    else if (x <= 4.5d-81) then
        tmp = t_3
    else if (x <= 7.2d+119) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double t_2 = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	double t_3 = -Math.log(y);
	double tmp;
	if (x <= -2.95e+63) {
		tmp = t_1;
	} else if (x <= 2e-246) {
		tmp = t_2;
	} else if (x <= 2.1e-228) {
		tmp = t_3;
	} else if (x <= 2.15e-99) {
		tmp = (y - (z * y)) - t;
	} else if (x <= 4.5e-81) {
		tmp = t_3;
	} else if (x <= 7.2e+119) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t
	t_3 = -math.log(y)
	tmp = 0
	if x <= -2.95e+63:
		tmp = t_1
	elif x <= 2e-246:
		tmp = t_2
	elif x <= 2.1e-228:
		tmp = t_3
	elif x <= 2.15e-99:
		tmp = (y - (z * y)) - t
	elif x <= 4.5e-81:
		tmp = t_3
	elif x <= 7.2e+119:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(Float64(Float64(y * y) * Float64(0.5 + Float64(z * -0.5))) - Float64(y * Float64(z + -1.0))) - t)
	t_3 = Float64(-log(y))
	tmp = 0.0
	if (x <= -2.95e+63)
		tmp = t_1;
	elseif (x <= 2e-246)
		tmp = t_2;
	elseif (x <= 2.1e-228)
		tmp = t_3;
	elseif (x <= 2.15e-99)
		tmp = Float64(Float64(y - Float64(z * y)) - t);
	elseif (x <= 4.5e-81)
		tmp = t_3;
	elseif (x <= 7.2e+119)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	t_3 = -log(y);
	tmp = 0.0;
	if (x <= -2.95e+63)
		tmp = t_1;
	elseif (x <= 2e-246)
		tmp = t_2;
	elseif (x <= 2.1e-228)
		tmp = t_3;
	elseif (x <= 2.15e-99)
		tmp = (y - (z * y)) - t;
	elseif (x <= 4.5e-81)
		tmp = t_3;
	elseif (x <= 7.2e+119)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(y * y), $MachinePrecision] * N[(0.5 + N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$3 = (-N[Log[y], $MachinePrecision])}, If[LessEqual[x, -2.95e+63], t$95$1, If[LessEqual[x, 2e-246], t$95$2, If[LessEqual[x, 2.1e-228], t$95$3, If[LessEqual[x, 2.15e-99], N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 4.5e-81], t$95$3, If[LessEqual[x, 7.2e+119], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.95000000000000014e63 or 7.20000000000000003e119 < x

   1. Initial program 95.7%

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

      1. associate--l+ 95.7%

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

      2. +-commutative 95.7%

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

      3. associate-+l- 95.7%

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

      4. *-commutative 95.7%

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

      5. *-commutative 95.7%

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

      6. sub-neg 95.7%

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

      7. metadata-eval 95.7%

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

      8. sub-neg 95.7%

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

      9. log1p-def 99.7%

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

      10. sub-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. *-commutative 99.7%

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

      13. distribute-rgt-neg-in 99.7%

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

      14. fma-def 99.7%

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

      15. neg-sub0 99.7%

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

      16. associate--r- 99.7%

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

      17. neg-sub0 99.7%

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

      18. +-commutative 99.7%

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

      19. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 76.1%

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

   if -2.95000000000000014e63 < x < 1.99999999999999991e-246 or 4.5e-81 < x < 7.20000000000000003e119

   1. Initial program 87.8%

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

      1. associate--l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+l- 87.8%

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

      4. *-commutative 87.8%

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

      5. *-commutative 87.8%

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

      6. sub-neg 87.8%

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

      7. metadata-eval 87.8%

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

      8. sub-neg 87.8%

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

      9. log1p-def 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. fma-def 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. neg-sub0 100.0%

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

      18. +-commutative 100.0%

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

      19. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 67.1%

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

   5. Taylor expanded in y around 0 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unsub-neg 67.1%

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

      3. associate-*r* 67.1%

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

      4. sub-neg 67.1%

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

      5. metadata-eval 67.1%

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

      6. +-commutative 67.1%

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

      7. distribute-lft-in 67.1%

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

      8. metadata-eval 67.1%

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

      9. unpow2 67.1%

         \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \color{blue}{\left(y \cdot y\right)} - \left(z - 1\right) \cdot y\right) - t \]

      10. *-commutative 67.1%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - \color{blue}{y \cdot \left(z - 1\right)}\right) - t \]

      11. sub-neg 67.1%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \color{blue}{\left(z + \left(-1\right)\right)}\right) - t \]

      12. metadata-eval 67.1%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \left(z + \color{blue}{-1}\right)\right) - t \]

      13. +-commutative 67.1%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \color{blue}{\left(-1 + z\right)}\right) - t \]

   7. Simplified 67.1%

      \[\leadsto \color{blue}{\left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \left(-1 + z\right)\right)} - t \]

   if 1.99999999999999991e-246 < x < 2.09999999999999991e-228 or 2.1499999999999999e-99 < x < 4.5e-81

   1. Initial program 97.3%

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

      1. associate--l+ 97.3%

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

      2. +-commutative 97.3%

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

      3. associate-+l- 97.3%

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

      4. *-commutative 97.3%

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

      5. *-commutative 97.3%

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

      6. sub-neg 97.3%

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

      7. metadata-eval 97.3%

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

      8. sub-neg 97.3%

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

      9. log1p-def 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. fma-def 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. neg-sub0 100.0%

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

      18. +-commutative 100.0%

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

      19. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. mul-1-neg 97.3%

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

      2. +-commutative 97.3%

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

      3. fma-def 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in t around 0 90.7%

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

   8. Taylor expanded in x around 0 90.7%

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

   if 2.09999999999999991e-228 < x < 2.1499999999999999e-99

   1. Initial program 81.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around inf 73.8%

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

       1. sub-neg 73.8%

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

       2. distribute-lft-in 73.8%

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

       3. *-rgt-identity 73.8%

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

       4. distribute-rgt-neg-in 73.8%

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

       5. unsub-neg 73.8%

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

   11. Simplified 73.8%

       \[\leadsto \color{blue}{\left(y - y \cdot z\right)} - t \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+63}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-246}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-228}:\\ \;\;\;\;-\log y\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-99}:\\ \;\;\;\;\left(y - z \cdot y\right) - t\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-81}:\\ \;\;\;\;-\log y\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]

Alternative 6: 95.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(-1 + x\right)\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-12} \lor \neg \left(t \leq 7.2 \cdot 10^{+16}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) (+ -1.0 x))))
   (if (or (<= t -8.5e-12) (not (<= t 7.2e+16)))
     (- t_1 t)
     (+ (* y (- 1.0 z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * (-1.0 + x);
	double tmp;
	if ((t <= -8.5e-12) || !(t <= 7.2e+16)) {
		tmp = t_1 - t;
	} else {
		tmp = (y * (1.0 - z)) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * ((-1.0d0) + x)
    if ((t <= (-8.5d-12)) .or. (.not. (t <= 7.2d+16))) then
        tmp = t_1 - t
    else
        tmp = (y * (1.0d0 - z)) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * (-1.0 + x);
	double tmp;
	if ((t <= -8.5e-12) || !(t <= 7.2e+16)) {
		tmp = t_1 - t;
	} else {
		tmp = (y * (1.0 - z)) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * (-1.0 + x)
	tmp = 0
	if (t <= -8.5e-12) or not (t <= 7.2e+16):
		tmp = t_1 - t
	else:
		tmp = (y * (1.0 - z)) + t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * Float64(-1.0 + x))
	tmp = 0.0
	if ((t <= -8.5e-12) || !(t <= 7.2e+16))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(Float64(y * Float64(1.0 - z)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * (-1.0 + x);
	tmp = 0.0;
	if ((t <= -8.5e-12) || ~((t <= 7.2e+16)))
		tmp = t_1 - t;
	else
		tmp = (y * (1.0 - z)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -8.5e-12], N[Not[LessEqual[t, 7.2e+16]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.4999999999999997e-12 or 7.2e16 < t

   1. Initial program 93.6%

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

   2. Taylor expanded in y around 0 93.3%

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

   if -8.4999999999999997e-12 < t < 7.2e16

   1. Initial program 87.0%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in y around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-commutative 98.9%

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

      5. distribute-rgt-neg-in 98.9%

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

      6. +-commutative 98.9%

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

      7. distribute-neg-in 98.9%

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

      8. metadata-eval 98.9%

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

      9. unsub-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in t around 0 98.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-12} \lor \neg \left(t \leq 7.2 \cdot 10^{+16}\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) + \log y \cdot \left(-1 + x\right)\\ \end{array} \]

Alternative 7: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-58}:\\ \;\;\;\;\left(y - z \cdot y\right) - t\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+32}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+63} \lor \neg \left(x \leq 3.1 \cdot 10^{+115}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -2.75e+63)
     t_1
     (if (<= x -4.5e-58)
       (- (- y (* z y)) t)
       (if (<= x 3.2e+32)
         (- (- (log y)) t)
         (if (or (<= x 1.95e+63) (not (<= x 3.1e+115)))
           t_1
           (- (* y (- z)) t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -2.75e+63) {
		tmp = t_1;
	} else if (x <= -4.5e-58) {
		tmp = (y - (z * y)) - t;
	} else if (x <= 3.2e+32) {
		tmp = -log(y) - t;
	} else if ((x <= 1.95e+63) || !(x <= 3.1e+115)) {
		tmp = t_1;
	} else {
		tmp = (y * -z) - 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 = log(y) * x
    if (x <= (-2.75d+63)) then
        tmp = t_1
    else if (x <= (-4.5d-58)) then
        tmp = (y - (z * y)) - t
    else if (x <= 3.2d+32) then
        tmp = -log(y) - t
    else if ((x <= 1.95d+63) .or. (.not. (x <= 3.1d+115))) then
        tmp = t_1
    else
        tmp = (y * -z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -2.75e+63) {
		tmp = t_1;
	} else if (x <= -4.5e-58) {
		tmp = (y - (z * y)) - t;
	} else if (x <= 3.2e+32) {
		tmp = -Math.log(y) - t;
	} else if ((x <= 1.95e+63) || !(x <= 3.1e+115)) {
		tmp = t_1;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -2.75e+63:
		tmp = t_1
	elif x <= -4.5e-58:
		tmp = (y - (z * y)) - t
	elif x <= 3.2e+32:
		tmp = -math.log(y) - t
	elif (x <= 1.95e+63) or not (x <= 3.1e+115):
		tmp = t_1
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -2.75e+63)
		tmp = t_1;
	elseif (x <= -4.5e-58)
		tmp = Float64(Float64(y - Float64(z * y)) - t);
	elseif (x <= 3.2e+32)
		tmp = Float64(Float64(-log(y)) - t);
	elseif ((x <= 1.95e+63) || !(x <= 3.1e+115))
		tmp = t_1;
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -2.75e+63)
		tmp = t_1;
	elseif (x <= -4.5e-58)
		tmp = (y - (z * y)) - t;
	elseif (x <= 3.2e+32)
		tmp = -log(y) - t;
	elseif ((x <= 1.95e+63) || ~((x <= 3.1e+115)))
		tmp = t_1;
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.75e+63], t$95$1, If[LessEqual[x, -4.5e-58], N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 3.2e+32], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], If[Or[LessEqual[x, 1.95e+63], N[Not[LessEqual[x, 3.1e+115]], $MachinePrecision]], t$95$1, N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.75000000000000002e63 or 3.1999999999999999e32 < x < 1.95e63 or 3.10000000000000005e115 < x

   1. Initial program 95.0%

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

      1. associate--l+ 95.0%

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

      2. +-commutative 95.0%

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

      3. associate-+l- 95.0%

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

      4. *-commutative 95.0%

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

      5. *-commutative 95.0%

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

      6. sub-neg 95.0%

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

      7. metadata-eval 95.0%

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

      8. sub-neg 95.0%

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

      9. log1p-def 99.7%

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

      10. sub-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. *-commutative 99.7%

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

      13. distribute-rgt-neg-in 99.7%

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

      14. fma-def 99.7%

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

      15. neg-sub0 99.7%

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

      16. associate--r- 99.7%

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

      17. neg-sub0 99.7%

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

      18. +-commutative 99.7%

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

      19. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 75.8%

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

   if -2.75000000000000002e63 < x < -4.5000000000000003e-58

   1. Initial program 77.0%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

   8. Simplified 88.8%

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

   9. Taylor expanded in y around inf 84.1%

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

       1. sub-neg 84.1%

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

       2. distribute-lft-in 84.1%

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

       3. *-rgt-identity 84.1%

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

       4. distribute-rgt-neg-in 84.1%

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

       5. unsub-neg 84.1%

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

   11. Simplified 84.1%

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

   if -4.5000000000000003e-58 < x < 3.1999999999999999e32

   1. Initial program 90.1%

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

      1. associate--l+ 90.1%

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

      2. +-commutative 90.1%

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

      3. associate-+l- 90.1%

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

      4. *-commutative 90.1%

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

      5. *-commutative 90.1%

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

      6. sub-neg 90.1%

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

      7. metadata-eval 90.1%

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

      8. sub-neg 90.1%

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

      9. log1p-def 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. fma-def 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. neg-sub0 100.0%

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

      18. +-commutative 100.0%

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

      19. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 89.2%

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

      1. mul-1-neg 89.2%

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

      2. +-commutative 89.2%

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

      3. fma-def 89.2%

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

   6. Simplified 89.2%

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

   7. Taylor expanded in x around 0 86.0%

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

      1. +-commutative 86.0%

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

   9. Simplified 86.0%

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

   if 1.95e63 < x < 3.10000000000000005e115

   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. associate--l+ 91.7%

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

      2. +-commutative 91.7%

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

      3. associate-+l- 91.7%

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

      4. *-commutative 91.7%

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

      5. *-commutative 91.7%

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

      6. sub-neg 91.7%

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

      7. metadata-eval 91.7%

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

      8. sub-neg 91.7%

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

      9. log1p-def 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. fma-def 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. neg-sub0 100.0%

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

      18. +-commutative 100.0%

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

      19. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in y around 0 76.2%

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

      1. mul-1-neg 76.2%

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

      2. distribute-rgt-neg-in 76.2%

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

      3. sub-neg 76.2%

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

      4. metadata-eval 76.2%

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

      5. +-commutative 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in z around inf 76.2%

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

      1. associate-*r* 76.2%

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

      2. mul-1-neg 76.2%

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

   10. Simplified 76.2%

       \[\leadsto \color{blue}{\left(-y\right) \cdot z} - t \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+63}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-58}:\\ \;\;\;\;\left(y - z \cdot y\right) - t\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+32}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+63} \lor \neg \left(x \leq 3.1 \cdot 10^{+115}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 8: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-58}:\\ \;\;\;\;\left(y - z \cdot y\right) - t\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+32}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+61} \lor \neg \left(x \leq 7.2 \cdot 10^{+119}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -1.9e+63)
     t_1
     (if (<= x -3.1e-58)
       (- (- y (* z y)) t)
       (if (<= x 3.1e+32)
         (- (- y (log y)) t)
         (if (or (<= x 6.2e+61) (not (<= x 7.2e+119)))
           t_1
           (- (* y (- z)) t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -1.9e+63) {
		tmp = t_1;
	} else if (x <= -3.1e-58) {
		tmp = (y - (z * y)) - t;
	} else if (x <= 3.1e+32) {
		tmp = (y - log(y)) - t;
	} else if ((x <= 6.2e+61) || !(x <= 7.2e+119)) {
		tmp = t_1;
	} else {
		tmp = (y * -z) - 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 = log(y) * x
    if (x <= (-1.9d+63)) then
        tmp = t_1
    else if (x <= (-3.1d-58)) then
        tmp = (y - (z * y)) - t
    else if (x <= 3.1d+32) then
        tmp = (y - log(y)) - t
    else if ((x <= 6.2d+61) .or. (.not. (x <= 7.2d+119))) then
        tmp = t_1
    else
        tmp = (y * -z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -1.9e+63) {
		tmp = t_1;
	} else if (x <= -3.1e-58) {
		tmp = (y - (z * y)) - t;
	} else if (x <= 3.1e+32) {
		tmp = (y - Math.log(y)) - t;
	} else if ((x <= 6.2e+61) || !(x <= 7.2e+119)) {
		tmp = t_1;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -1.9e+63:
		tmp = t_1
	elif x <= -3.1e-58:
		tmp = (y - (z * y)) - t
	elif x <= 3.1e+32:
		tmp = (y - math.log(y)) - t
	elif (x <= 6.2e+61) or not (x <= 7.2e+119):
		tmp = t_1
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -1.9e+63)
		tmp = t_1;
	elseif (x <= -3.1e-58)
		tmp = Float64(Float64(y - Float64(z * y)) - t);
	elseif (x <= 3.1e+32)
		tmp = Float64(Float64(y - log(y)) - t);
	elseif ((x <= 6.2e+61) || !(x <= 7.2e+119))
		tmp = t_1;
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -1.9e+63)
		tmp = t_1;
	elseif (x <= -3.1e-58)
		tmp = (y - (z * y)) - t;
	elseif (x <= 3.1e+32)
		tmp = (y - log(y)) - t;
	elseif ((x <= 6.2e+61) || ~((x <= 7.2e+119)))
		tmp = t_1;
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.9e+63], t$95$1, If[LessEqual[x, -3.1e-58], N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 3.1e+32], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[x, 6.2e+61], N[Not[LessEqual[x, 7.2e+119]], $MachinePrecision]], t$95$1, N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.9000000000000001e63 or 3.09999999999999993e32 < x < 6.1999999999999998e61 or 7.20000000000000003e119 < x

   1. Initial program 95.0%

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

      1. associate--l+ 95.0%

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

      2. +-commutative 95.0%

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

      3. associate-+l- 95.0%

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

      4. *-commutative 95.0%

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

      5. *-commutative 95.0%

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

      6. sub-neg 95.0%

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

      7. metadata-eval 95.0%

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

      8. sub-neg 95.0%

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

      9. log1p-def 99.7%

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

      10. sub-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. *-commutative 99.7%

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

      13. distribute-rgt-neg-in 99.7%

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

      14. fma-def 99.7%

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

      15. neg-sub0 99.7%

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

      16. associate--r- 99.7%

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

      17. neg-sub0 99.7%

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

      18. +-commutative 99.7%

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

      19. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 75.8%

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

   if -1.9000000000000001e63 < x < -3.0999999999999999e-58

   1. Initial program 77.0%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

   8. Simplified 88.8%

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

   9. Taylor expanded in y around inf 84.1%

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

       1. sub-neg 84.1%

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

       2. distribute-lft-in 84.1%

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

       3. *-rgt-identity 84.1%

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

       4. distribute-rgt-neg-in 84.1%

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

       5. unsub-neg 84.1%

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

   11. Simplified 84.1%

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

   if -3.0999999999999999e-58 < x < 3.09999999999999993e32

   1. Initial program 90.1%

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

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-neg-in 99.5%

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

      8. metadata-eval 99.5%

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

      9. unsub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 96.2%

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

      1. mul-1-neg 96.2%

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

      2. unsub-neg 96.2%

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

   8. Simplified 96.2%

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

   9. Taylor expanded in z around 0 86.1%

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

   if 6.1999999999999998e61 < x < 7.20000000000000003e119

   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. associate--l+ 91.7%

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

      2. +-commutative 91.7%

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

      3. associate-+l- 91.7%

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

      4. *-commutative 91.7%

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

      5. *-commutative 91.7%

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

      6. sub-neg 91.7%

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

      7. metadata-eval 91.7%

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

      8. sub-neg 91.7%

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

      9. log1p-def 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. fma-def 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. neg-sub0 100.0%

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

      18. +-commutative 100.0%

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

      19. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in y around 0 76.2%

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

      1. mul-1-neg 76.2%

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

      2. distribute-rgt-neg-in 76.2%

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

      3. sub-neg 76.2%

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

      4. metadata-eval 76.2%

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

      5. +-commutative 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in z around inf 76.2%

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

      1. associate-*r* 76.2%

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

      2. mul-1-neg 76.2%

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

   10. Simplified 76.2%

       \[\leadsto \color{blue}{\left(-y\right) \cdot z} - t \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+63}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-58}:\\ \;\;\;\;\left(y - z \cdot y\right) - t\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+32}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+61} \lor \neg \left(x \leq 7.2 \cdot 10^{+119}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 9: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in y around 0 99.9%

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

3. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-commutative 99.4%

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

   5. distribute-rgt-neg-in 99.4%

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

   6. +-commutative 99.4%

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

   7. distribute-neg-in 99.4%

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

   8. metadata-eval 99.4%

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

   9. unsub-neg 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 10: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in y around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. unsub-neg 99.8%

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

   3. *-commutative 99.8%

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

   4. unpow2 99.8%

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

   5. associate-*l* 99.8%

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

4. Simplified 99.8%

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

5. Taylor expanded in z around inf 99.8%

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

   1. unpow2 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in y around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-commutative 99.4%

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

   5. /-rgt-identity 99.4%

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

   6. associate-/r/ 99.3%

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

   7. mul-1-neg 99.3%

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

   8. unsub-neg 99.3%

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

   9. associate-/r/ 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. +-commutative 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 11: 78.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -13200000000:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -13200000000.0)
   (- (- (* (* y y) (+ 0.5 (* z -0.5))) (* y (+ z -1.0))) t)
   (if (<= t 9.5e+62) (* (log y) (+ -1.0 x)) (- (* y (- z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -13200000000.0) {
		tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	} else if (t <= 9.5e+62) {
		tmp = log(y) * (-1.0 + x);
	} else {
		tmp = (y * -z) - 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 <= (-13200000000.0d0)) then
        tmp = (((y * y) * (0.5d0 + (z * (-0.5d0)))) - (y * (z + (-1.0d0)))) - t
    else if (t <= 9.5d+62) then
        tmp = log(y) * ((-1.0d0) + x)
    else
        tmp = (y * -z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -13200000000.0) {
		tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	} else if (t <= 9.5e+62) {
		tmp = Math.log(y) * (-1.0 + x);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -13200000000.0:
		tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t
	elif t <= 9.5e+62:
		tmp = math.log(y) * (-1.0 + x)
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -13200000000.0)
		tmp = Float64(Float64(Float64(Float64(y * y) * Float64(0.5 + Float64(z * -0.5))) - Float64(y * Float64(z + -1.0))) - t);
	elseif (t <= 9.5e+62)
		tmp = Float64(log(y) * Float64(-1.0 + x));
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -13200000000.0)
		tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	elseif (t <= 9.5e+62)
		tmp = log(y) * (-1.0 + x);
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -13200000000.0], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(0.5 + N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 9.5e+62], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.32e10

   1. Initial program 93.5%

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

      1. associate--l+ 93.5%

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

      2. +-commutative 93.5%

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

      3. associate-+l- 93.5%

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

      4. *-commutative 93.5%

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

      5. *-commutative 93.5%

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

      6. sub-neg 93.5%

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

      7. metadata-eval 93.5%

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

      8. sub-neg 93.5%

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

      9. log1p-def 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. distribute-rgt-neg-in 99.9%

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

      14. fma-def 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 79.1%

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

   5. Taylor expanded in y around 0 79.1%

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

      1. mul-1-neg 79.1%

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

      2. unsub-neg 79.1%

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

      3. associate-*r* 79.1%

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

      4. sub-neg 79.1%

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

      5. metadata-eval 79.1%

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

      6. +-commutative 79.1%

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

      7. distribute-lft-in 79.1%

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

      8. metadata-eval 79.1%

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

      9. unpow2 79.1%

         \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \color{blue}{\left(y \cdot y\right)} - \left(z - 1\right) \cdot y\right) - t \]

      10. *-commutative 79.1%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - \color{blue}{y \cdot \left(z - 1\right)}\right) - t \]

      11. sub-neg 79.1%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \color{blue}{\left(z + \left(-1\right)\right)}\right) - t \]

      12. metadata-eval 79.1%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \left(z + \color{blue}{-1}\right)\right) - t \]

      13. +-commutative 79.1%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \color{blue}{\left(-1 + z\right)}\right) - t \]

   7. Simplified 79.1%

      \[\leadsto \color{blue}{\left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \left(-1 + z\right)\right)} - t \]

   if -1.32e10 < t < 9.5000000000000003e62

   1. Initial program 87.8%

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

      1. associate--l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+l- 87.8%

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

      4. *-commutative 87.8%

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

      5. *-commutative 87.8%

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

      6. sub-neg 87.8%

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

      7. metadata-eval 87.8%

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

      8. sub-neg 87.8%

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

      9. log1p-def 99.8%

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

      10. sub-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. *-commutative 99.8%

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

      13. distribute-rgt-neg-in 99.8%

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

      14. fma-def 99.8%

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

      15. neg-sub0 99.8%

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

      16. associate--r- 99.8%

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

      17. neg-sub0 99.8%

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

      18. +-commutative 99.8%

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

      19. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 86.4%

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

      1. mul-1-neg 86.4%

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

      2. +-commutative 86.4%

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

      3. fma-def 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in t around 0 83.3%

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

   if 9.5000000000000003e62 < t

   1. Initial program 93.6%

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

      1. associate--l+ 93.6%

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

      2. +-commutative 93.6%

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

      3. associate-+l- 93.6%

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

      4. *-commutative 93.6%

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

      5. *-commutative 93.6%

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

      6. sub-neg 93.6%

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

      7. metadata-eval 93.6%

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

      8. sub-neg 93.6%

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

      9. log1p-def 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. distribute-rgt-neg-in 99.9%

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

      14. fma-def 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 81.6%

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

   5. Taylor expanded in y around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. distribute-rgt-neg-in 81.6%

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

      3. sub-neg 81.6%

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

      4. metadata-eval 81.6%

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

      5. +-commutative 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in z around inf 81.6%

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

      1. associate-*r* 81.6%

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

      2. mul-1-neg 81.6%

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

   10. Simplified 81.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -13200000000:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 12: 88.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in y around 0 89.7%

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

3. Final simplification 89.7%

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

Alternative 13: 56.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00075 \lor \neg \left(t \leq 2.05 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -0.00075) (not (<= t 2.05e-8)))
   (- (- (* (* y y) (+ 0.5 (* z -0.5))) (* y (+ z -1.0))) t)
   (- (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00075) || !(t <= 2.05e-8)) {
		tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	} else {
		tmp = -log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.00075d0)) .or. (.not. (t <= 2.05d-8))) then
        tmp = (((y * y) * (0.5d0 + (z * (-0.5d0)))) - (y * (z + (-1.0d0)))) - t
    else
        tmp = -log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00075) || !(t <= 2.05e-8)) {
		tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	} else {
		tmp = -Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -0.00075) or not (t <= 2.05e-8):
		tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t
	else:
		tmp = -math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -0.00075) || !(t <= 2.05e-8))
		tmp = Float64(Float64(Float64(Float64(y * y) * Float64(0.5 + Float64(z * -0.5))) - Float64(y * Float64(z + -1.0))) - t);
	else
		tmp = Float64(-log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -0.00075) || ~((t <= 2.05e-8)))
		tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
	else
		tmp = -log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -0.00075], N[Not[LessEqual[t, 2.05e-8]], $MachinePrecision]], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(0.5 + N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[Log[y], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if t < -7.5000000000000002e-4 or 2.05000000000000016e-8 < t

   1. Initial program 92.6%

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

      1. associate--l+ 92.6%

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

      2. +-commutative 92.6%

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

      3. associate-+l- 92.6%

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

      4. *-commutative 92.6%

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

      5. *-commutative 92.6%

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

      6. sub-neg 92.6%

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

      7. metadata-eval 92.6%

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

      8. sub-neg 92.6%

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

      9. log1p-def 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. distribute-rgt-neg-in 99.9%

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

      14. fma-def 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 76.0%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. mul-1-neg 76.0%

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

      2. unsub-neg 76.0%

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

      3. associate-*r* 76.0%

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

      4. sub-neg 76.0%

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

      5. metadata-eval 76.0%

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

      6. +-commutative 76.0%

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

      7. distribute-lft-in 76.0%

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

      8. metadata-eval 76.0%

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

      9. unpow2 76.0%

         \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \color{blue}{\left(y \cdot y\right)} - \left(z - 1\right) \cdot y\right) - t \]

      10. *-commutative 76.0%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - \color{blue}{y \cdot \left(z - 1\right)}\right) - t \]

      11. sub-neg 76.0%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \color{blue}{\left(z + \left(-1\right)\right)}\right) - t \]

      12. metadata-eval 76.0%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \left(z + \color{blue}{-1}\right)\right) - t \]

      13. +-commutative 76.0%

          \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \color{blue}{\left(-1 + z\right)}\right) - t \]

   7. Simplified 76.0%

      \[\leadsto \color{blue}{\left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \left(-1 + z\right)\right)} - t \]

   if -7.5000000000000002e-4 < t < 2.05000000000000016e-8

   1. Initial program 88.2%

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

      1. associate--l+ 88.2%

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

      2. +-commutative 88.2%

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

      3. associate-+l- 88.2%

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

      4. *-commutative 88.2%

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

      5. *-commutative 88.2%

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

      6. sub-neg 88.2%

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

      7. metadata-eval 88.2%

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

      8. sub-neg 88.2%

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

      9. log1p-def 99.8%

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

      10. sub-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. *-commutative 99.8%

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

      13. distribute-rgt-neg-in 99.8%

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

      14. fma-def 99.8%

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

      15. neg-sub0 99.8%

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

      16. associate--r- 99.8%

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

      17. neg-sub0 99.8%

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

      18. +-commutative 99.8%

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

      19. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. +-commutative 87.2%

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

      3. fma-def 87.2%

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

   6. Simplified 87.2%

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

   7. Taylor expanded in t around 0 86.2%

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

   8. Taylor expanded in x around 0 40.5%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00075 \lor \neg \left(t \leq 2.05 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \]

Alternative 14: 47.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (- (* (* y y) (+ 0.5 (* z -0.5))) (* y (+ z -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (((y * y) * (0.5 + (z * -0.5))) - (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 = (((y * y) * (0.5d0 + (z * (-0.5d0)))) - (y * (z + (-1.0d0)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
}

Python

def code(x, y, z, t):
	return (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(y * y) * Float64(0.5 + Float64(z * -0.5))) - Float64(y * Float64(z + -1.0))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((y * y) * (0.5 + (z * -0.5))) - (y * (z + -1.0))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(y * y), $MachinePrecision] * N[(0.5 + N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 90.6%

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

   1. associate--l+ 90.6%

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

   2. +-commutative 90.6%

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

   3. associate-+l- 90.6%

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

   4. *-commutative 90.6%

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

   5. *-commutative 90.6%

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

   6. sub-neg 90.6%

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

   7. metadata-eval 90.6%

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

   8. sub-neg 90.6%

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

   9. log1p-def 99.9%

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

   10. sub-neg 99.9%

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

   11. +-commutative 99.9%

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

   12. *-commutative 99.9%

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

   13. distribute-rgt-neg-in 99.9%

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

   14. fma-def 99.9%

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

   15. neg-sub0 99.9%

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

   16. associate--r- 99.9%

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

   17. neg-sub0 99.9%

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

   18. +-commutative 99.9%

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

   19. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around inf 48.0%

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

5. Taylor expanded in y around 0 48.0%

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

   1. mul-1-neg 48.0%

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

   2. unsub-neg 48.0%

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

   3. associate-*r* 48.0%

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

   4. sub-neg 48.0%

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

   5. metadata-eval 48.0%

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

   6. +-commutative 48.0%

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

   7. distribute-lft-in 48.0%

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

   8. metadata-eval 48.0%

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

   9. unpow2 48.0%

      \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \color{blue}{\left(y \cdot y\right)} - \left(z - 1\right) \cdot y\right) - t \]

   10. *-commutative 48.0%

       \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - \color{blue}{y \cdot \left(z - 1\right)}\right) - t \]

   11. sub-neg 48.0%

       \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \color{blue}{\left(z + \left(-1\right)\right)}\right) - t \]

   12. metadata-eval 48.0%

       \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \left(z + \color{blue}{-1}\right)\right) - t \]

   13. +-commutative 48.0%

       \[\leadsto \left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \color{blue}{\left(-1 + z\right)}\right) - t \]

7. Simplified 48.0%

   \[\leadsto \color{blue}{\left(\left(0.5 + -0.5 \cdot z\right) \cdot \left(y \cdot y\right) - y \cdot \left(-1 + z\right)\right)} - t \]

8. Final simplification 48.0%

   \[\leadsto \left(\left(y \cdot y\right) \cdot \left(0.5 + z \cdot -0.5\right) - y \cdot \left(z + -1\right)\right) - t \]

Alternative 15: 46.8% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in y around 0 99.9%

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

3. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-commutative 99.4%

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

   5. distribute-rgt-neg-in 99.4%

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

   6. +-commutative 99.4%

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

   7. distribute-neg-in 99.4%

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

   8. metadata-eval 99.4%

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

   9. unsub-neg 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0 65.4%

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

   1. mul-1-neg 65.4%

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

   2. unsub-neg 65.4%

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

8. Simplified 65.4%

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

9. Taylor expanded in y around inf 47.7%

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

    1. sub-neg 47.7%

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

    2. distribute-lft-in 47.7%

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

    3. *-rgt-identity 47.7%

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

    4. distribute-rgt-neg-in 47.7%

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

    5. unsub-neg 47.7%

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

11. Simplified 47.7%

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

12. Final simplification 47.7%

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

Alternative 16: 46.6% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate--l+ 90.6%

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

   2. +-commutative 90.6%

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

   3. associate-+l- 90.6%

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

   4. *-commutative 90.6%

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

   5. *-commutative 90.6%

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

   6. sub-neg 90.6%

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

   7. metadata-eval 90.6%

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

   8. sub-neg 90.6%

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

   9. log1p-def 99.9%

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

   10. sub-neg 99.9%

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

   11. +-commutative 99.9%

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

   12. *-commutative 99.9%

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

   13. distribute-rgt-neg-in 99.9%

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

   14. fma-def 99.9%

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

   15. neg-sub0 99.9%

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

   16. associate--r- 99.9%

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

   17. neg-sub0 99.9%

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

   18. +-commutative 99.9%

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

   19. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around inf 48.0%

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

5. Taylor expanded in y around 0 47.7%

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

   1. mul-1-neg 47.7%

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

   2. distribute-rgt-neg-in 47.7%

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

   3. sub-neg 47.7%

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

   4. metadata-eval 47.7%

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

   5. +-commutative 47.7%

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

7. Simplified 47.7%

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

8. Taylor expanded in z around inf 47.5%

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

   1. associate-*r* 47.5%

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

   2. mul-1-neg 47.5%

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

10. Simplified 47.5%

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

11. Final simplification 47.5%

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

Alternative 17: 36.7% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate--l+ 90.6%

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

   2. +-commutative 90.6%

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

   3. associate-+l- 90.6%

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

   4. *-commutative 90.6%

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

   5. *-commutative 90.6%

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

   6. sub-neg 90.6%

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

   7. metadata-eval 90.6%

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

   8. sub-neg 90.6%

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

   9. log1p-def 99.9%

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

   10. sub-neg 99.9%

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

   11. +-commutative 99.9%

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

   12. *-commutative 99.9%

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

   13. distribute-rgt-neg-in 99.9%

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

   14. fma-def 99.9%

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

   15. neg-sub0 99.9%

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

   16. associate--r- 99.9%

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

   17. neg-sub0 99.9%

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

   18. +-commutative 99.9%

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

   19. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around inf 48.0%

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

5. Taylor expanded in y around 0 47.7%

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

   1. mul-1-neg 47.7%

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

   2. distribute-rgt-neg-in 47.7%

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

   3. sub-neg 47.7%

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

   4. metadata-eval 47.7%

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

   5. +-commutative 47.7%

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

7. Simplified 47.7%

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

8. Taylor expanded in z around 0 39.0%

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

9. Final simplification 39.0%

   \[\leadsto y - t \]

Alternative 18: 36.4% accurate, 107.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate--l+ 90.6%

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

   2. +-commutative 90.6%

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

   3. associate-+l- 90.6%

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

   4. *-commutative 90.6%

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

   5. *-commutative 90.6%

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

   6. sub-neg 90.6%

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

   7. metadata-eval 90.6%

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

   8. sub-neg 90.6%

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

   9. log1p-def 99.9%

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

   10. sub-neg 99.9%

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

   11. +-commutative 99.9%

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

   12. *-commutative 99.9%

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

   13. distribute-rgt-neg-in 99.9%

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

   14. fma-def 99.9%

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

   15. neg-sub0 99.9%

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

   16. associate--r- 99.9%

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

   17. neg-sub0 99.9%

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

   18. +-commutative 99.9%

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

   19. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around inf 38.7%

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

   1. neg-mul-1 38.7%

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

6. Simplified 38.7%

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

7. Final simplification 38.7%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023214 
(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))