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

Percentage Accurate: 89.0% → 99.8%

Time: 18.5s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. cancel-sign-sub 90.9%

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

   2. distribute-lft-neg-in 90.9%

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

   3. fma-neg 90.9%

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

   4. remove-double-neg 90.9%

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

   5. sub-neg 90.9%

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

   6. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((((y * (y * -0.5)) - y) * (z + -1.0)) + ((x + -1.0) * log(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 * (y * (-0.5d0))) - y) * (z + (-1.0d0))) + ((x + (-1.0d0)) * log(y))) - 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)) + ((x + -1.0) * Math.log(y))) - t;
}

Python

def code(x, y, z, t):
	return ((((y * (y * -0.5)) - y) * (z + -1.0)) + ((x + -1.0) * math.log(y))) - 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(Float64(x + -1.0) * log(y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((((y * (y * -0.5)) - y) * (z + -1.0)) + ((x + -1.0) * log(y))) - 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[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 90.9%

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

2. Taylor expanded in y around 0 99.2%

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

   1. mul-1-neg 99.2%

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

   2. unsub-neg 99.2%

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

   3. *-commutative 99.2%

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

   4. unpow2 99.2%

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

   5. associate-*l* 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 87.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 90.4%

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

   2. Taylor expanded in y around 0 88.2%

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

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

   1. Initial program 50.4%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-commutative 100.0%

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

      11. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 66.1%

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

      1. fma-neg 66.1%

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

      2. *-lft-identity 66.1%

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

      3. pow-base-1 66.1%

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

      4. mul-1-neg 66.1%

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

      5. fma-def 66.1%

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

      6. +-commutative 66.1%

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

      7. pow-base-1 66.1%

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

      8. *-lft-identity 66.1%

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

      9. mul-1-neg 66.1%

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

      10. unsub-neg 66.1%

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

      11. associate-*r* 66.1%

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

      12. sub-neg 66.1%

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

      13. metadata-eval 66.1%

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

      14. +-commutative 66.1%

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

      15. neg-mul-1 66.1%

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

      16. *-commutative 66.1%

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

      17. distribute-neg-in 66.1%

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

      18. metadata-eval 66.1%

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

      19. sub-neg 66.1%

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

   7. Simplified 66.1%

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

   if 1 < (-.f64 x 1)

   1. Initial program 93.3%

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

   2. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. mul-1-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. *-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. fma-def 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 92.8%

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

4. Final simplification 89.3%

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

Alternative 4: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

2. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. sub-neg 98.9%

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

   3. metadata-eval 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. +-commutative 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. *-commutative 98.9%

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

   11. +-commutative 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 5: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

2. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. sub-neg 98.9%

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

   3. metadata-eval 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. +-commutative 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. *-commutative 98.9%

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

   11. +-commutative 98.9%

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

4. Simplified 98.9%

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

5. Taylor expanded in z around inf 98.7%

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

6. Final simplification 98.7%

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

Alternative 6: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-96} \lor \neg \left(x \leq 5.5\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.8e-96) (not (<= x 5.5)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.8e-96) || !(x <= 5.5)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.8e-96) || !(x <= 5.5)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.8e-96) or not (x <= 5.5):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.8e-96) || !(x <= 5.5))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.8e-96], N[Not[LessEqual[x, 5.5]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8000000000000002e-96 or 5.5 < x

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. mul-1-neg 99.3%

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

      5. unsub-neg 99.3%

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

      6. *-commutative 99.3%

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

      7. +-commutative 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

      10. *-commutative 99.3%

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

      11. +-commutative 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in y around 0 99.3%

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

      1. fma-def 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 89.6%

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

   if -6.8000000000000002e-96 < x < 5.5

   1. Initial program 82.2%

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

   2. Taylor expanded in z around inf 47.2%

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

      1. sub-neg 47.2%

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

      2. mul-1-neg 47.2%

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

      3. log1p-def 63.8%

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

      4. mul-1-neg 63.8%

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

   4. Simplified 63.8%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-96} \lor \neg \left(x \leq 5.5\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 7: 76.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-96} \lor \neg \left(x \leq 1650\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(-0.5 \cdot \left(y \cdot y\right)\right) - t\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.8e-96) (not (<= x 1650.0)))
   (- (* x (log y)) t)
   (- (- (* z (* -0.5 (* y y))) t) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.8e-96) || !(x <= 1650.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = ((z * (-0.5 * (y * y))) - t) - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-6.8d-96)) .or. (.not. (x <= 1650.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = ((z * ((-0.5d0) * (y * y))) - t) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.8e-96) || !(x <= 1650.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = ((z * (-0.5 * (y * y))) - t) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.8e-96) or not (x <= 1650.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = ((z * (-0.5 * (y * y))) - t) - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.8e-96) || !(x <= 1650.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(z * Float64(-0.5 * Float64(y * y))) - t) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -6.8e-96) || ~((x <= 1650.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = ((z * (-0.5 * (y * y))) - t) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.8e-96], N[Not[LessEqual[x, 1650.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(z * N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8000000000000002e-96 or 1650 < x

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. mul-1-neg 99.3%

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

      5. unsub-neg 99.3%

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

      6. *-commutative 99.3%

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

      7. +-commutative 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

      10. *-commutative 99.3%

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

      11. +-commutative 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in y around 0 99.3%

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

      1. fma-def 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 89.6%

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

   if -6.8000000000000002e-96 < x < 1650

   1. Initial program 82.2%

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

   2. Taylor expanded in z around inf 47.2%

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

      1. sub-neg 47.2%

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

      2. mul-1-neg 47.2%

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

      3. log1p-def 63.8%

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

      4. mul-1-neg 63.8%

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

   4. Simplified 63.8%

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

   5. Taylor expanded in y around 0 62.5%

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

      1. neg-mul-1 62.5%

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

      2. associate-+r+ 62.5%

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

      3. mul-1-neg 62.5%

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

      4. *-commutative 62.5%

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

      5. unsub-neg 62.5%

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

      6. unsub-neg 62.5%

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

      7. associate-*r* 62.5%

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

      8. unpow2 62.5%

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

      9. *-commutative 62.5%

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

   7. Simplified 62.5%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left(y \cdot y\right)\right) \cdot z - t\right) - y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-96} \lor \neg \left(x \leq 1650\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(-0.5 \cdot \left(y \cdot y\right)\right) - t\right) - y \cdot z\\ \end{array} \]

Alternative 8: 88.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+260}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.6e+260) (- (* z (log1p (- y))) t) (- (* (+ x -1.0) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.6e+260) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = ((x + -1.0) * log(y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.6e+260) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.6e+260:
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = ((x + -1.0) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.6e+260)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.6e+260], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.60000000000000022e260

   1. Initial program 44.6%

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

   2. Taylor expanded in z around inf 28.0%

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

      1. sub-neg 28.0%

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

      2. mul-1-neg 28.0%

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

      3. log1p-def 83.2%

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

      4. mul-1-neg 83.2%

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

   4. Simplified 83.2%

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

   if -4.60000000000000022e260 < z

   1. Initial program 93.0%

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

   2. Taylor expanded in y around 0 91.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+260}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]

Alternative 9: 46.1% accurate, 16.5× speedup

Math

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

FPCore

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

C

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

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))) - t) - (y * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

2. Taylor expanded in z around inf 35.3%

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

   1. sub-neg 35.3%

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

   2. mul-1-neg 35.3%

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

   3. log1p-def 43.7%

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

   4. mul-1-neg 43.7%

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

4. Simplified 43.7%

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

5. Taylor expanded in y around 0 43.1%

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

   1. neg-mul-1 43.1%

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

   2. associate-+r+ 43.1%

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

   3. mul-1-neg 43.1%

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

   4. *-commutative 43.1%

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

   5. unsub-neg 43.1%

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

   6. unsub-neg 43.1%

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

   7. associate-*r* 43.1%

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

   8. unpow2 43.1%

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

   9. *-commutative 43.1%

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

7. Simplified 43.1%

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

8. Final simplification 43.1%

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

Alternative 10: 46.0% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

2. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. sub-neg 98.9%

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

   3. metadata-eval 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. +-commutative 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. *-commutative 98.9%

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

   11. +-commutative 98.9%

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

4. Simplified 98.9%

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

5. Taylor expanded in y around inf 43.0%

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

6. Final simplification 43.0%

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

Alternative 11: 45.8% accurate, 35.8× speedup

Math

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

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(-Float64(y * 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 90.9%

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

2. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. sub-neg 98.9%

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

   3. metadata-eval 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. +-commutative 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. *-commutative 98.9%

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

   11. +-commutative 98.9%

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

4. Simplified 98.9%

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

5. Taylor expanded in z around inf 42.8%

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

   1. associate-*r* 42.8%

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

   2. mul-1-neg 42.8%

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

7. Simplified 42.8%

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

8. Final simplification 42.8%

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

Alternative 12: 35.1% accurate, 43.0× speedup

Math

\[\begin{array}{l} \\ y \cdot z - t \end{array} \]

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 * 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 90.9%

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

2. Taylor expanded in z around inf 35.3%

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

   1. sub-neg 35.3%

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

   2. mul-1-neg 35.3%

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

   3. log1p-def 43.7%

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

   4. mul-1-neg 43.7%

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

4. Simplified 43.7%

   \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right)} - t \]
5. Step-by-step derivation

   1. sub-neg 43.7%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 33.6%

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

   4. sqr-neg 33.6%

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

   5. sqrt-unprod 33.7%

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

   6. add-sqr-sqrt 33.7%

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

6. Applied egg-rr 33.7%

   \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(y\right) + \left(-t\right)} \]
7. Step-by-step derivation

   1. sub-neg 33.7%

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

8. Simplified 33.7%

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

9. Taylor expanded in y around 0 33.7%

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

10. Final simplification 33.7%

    \[\leadsto y \cdot z - t \]

Alternative 13: 35.3% 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 90.9%

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

   1. cancel-sign-sub 90.9%

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

   2. distribute-lft-neg-in 90.9%

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

   3. fma-neg 90.9%

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

   4. remove-double-neg 90.9%

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

   5. sub-neg 90.9%

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

   6. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 33.7%

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

   1. neg-mul-1 33.7%

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

6. Simplified 33.7%

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

7. Final simplification 33.7%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023181 
(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))