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

Percentage Accurate: 89.0% → 99.8%

Time: 18.1s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. +-commutative 91.7%

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

   2. fma-define 91.7%

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

   3. sub-neg 91.7%

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

   4. metadata-eval 91.7%

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

   5. sub-neg 91.7%

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

   6. log1p-define 99.8%

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

   7. sub-neg 99.8%

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(y \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) \cdot \left(z + -1\right) + \log y \cdot \left(-1 + x\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (*
    (* y (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5))))
    (+ z -1.0))
   (* (log y) (+ -1.0 x)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)))) * (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 * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)))) * (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 * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)))) * (z + -1.0)) + (Math.log(y) * (-1.0 + x))) - t;
}

Python

def code(x, y, z, t):
	return (((y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)))) * (z + -1.0)) + (math.log(y) * (-1.0 + x))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5)))) * Float64(z + -1.0)) + Float64(log(y) * Float64(-1.0 + x))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)))) * (z + -1.0)) + (log(y) * (-1.0 + x))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(y * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 91.7%

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

3. Taylor expanded in y around 0 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}\right) - t \]

4. Final simplification 99.7%

   \[\leadsto \left(\left(y \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) \cdot \left(z + -1\right) + \log y \cdot \left(-1 + x\right)\right) - t \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.2 \lor \neg \left(-1 + x \leq -1\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -1.2) (not (<= (+ -1.0 x) -1.0)))
   (- (* (log y) (+ -1.0 x)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.2) || !((-1.0 + x) <= -1.0)) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-1.2d0)) .or. (.not. (((-1.0d0) + x) <= (-1.0d0)))) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.2) || !((-1.0 + x) <= -1.0)) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -1.2) or not ((-1.0 + x) <= -1.0):
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -1.2) || !(Float64(-1.0 + x) <= -1.0))
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -1.2) || ~(((-1.0 + x) <= -1.0)))
		tmp = (log(y) * (-1.0 + x)) - t;
	else
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.2], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0]], $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 #s(literal 1 binary64)) < -1.19999999999999996 or -1 < (-.f64 x #s(literal 1 binary64))

   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. Add Preprocessing

   3. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

      4. fma-define 99.0%

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

      5. mul-1-neg 99.0%

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

      6. fma-neg 99.0%

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

      7. +-commutative 99.0%

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

      8. sub-neg 99.0%

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

      9. metadata-eval 99.0%

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

      10. +-commutative 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in z around inf 78.2%

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

      1. sub-neg 78.2%

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

      2. metadata-eval 78.2%

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

      3. +-commutative 78.2%

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

      4. mul-1-neg 78.2%

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

      5. unsub-neg 78.2%

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

   8. Simplified 78.2%

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

   9. Taylor expanded in y around 0 94.0%

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

   if -1.19999999999999996 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. sub-neg 87.7%

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

      5. metadata-eval 87.7%

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

      6. +-commutative 87.7%

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

      7. sub-neg 87.7%

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

      8. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. sub-neg 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.2 \lor \neg \left(-1 + x \leq -1\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(y \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) \cdot \left(z + -1\right) + \log y \cdot \left(-1 + x\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (* y (+ -1.0 (* y (- (* y -0.3333333333333333) 0.5)))) (+ z -1.0))
   (* (log y) (+ -1.0 x)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))) * (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 * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0)))) * (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 * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))) * (z + -1.0)) + (Math.log(y) * (-1.0 + x))) - t;
}

Python

def code(x, y, z, t):
	return (((y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))) * (z + -1.0)) + (math.log(y) * (-1.0 + x))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5)))) * Float64(z + -1.0)) + Float64(log(y) * Float64(-1.0 + x))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))) * (z + -1.0)) + (log(y) * (-1.0 + x))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(y * N[(-1.0 + N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 91.7%

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

3. Taylor expanded in y around 0 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \left(\left(y \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) \cdot \left(z + -1\right) + \log y \cdot \left(-1 + x\right)\right) - t \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(y * N[(-1.0 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $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 91.7%

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

3. Taylor expanded in y around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 6: 87.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -53.0) (not (<= x 1.0))) (- (* x (log y)) t) (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -53.0) || !(x <= 1.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-53.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -53.0) || !(x <= 1.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -53.0) or not (x <= 1.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -53.0) || !(x <= 1.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -53.0) || ~((x <= 1.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -53.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -53 or 1 < x

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

      4. fma-define 99.0%

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

      5. mul-1-neg 99.0%

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

      6. fma-neg 99.0%

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

      7. +-commutative 99.0%

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

      8. sub-neg 99.0%

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

      9. metadata-eval 99.0%

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

      10. +-commutative 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in z around inf 77.1%

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

      1. sub-neg 77.1%

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

      2. metadata-eval 77.1%

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

      3. +-commutative 77.1%

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

      4. mul-1-neg 77.1%

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

      5. unsub-neg 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in x around inf 92.1%

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

       1. *-commutative 92.1%

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

   11. Simplified 92.1%

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

   if -53 < x < 1

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 86.8%

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

      1. +-commutative 86.8%

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

      2. mul-1-neg 86.8%

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

      3. unsub-neg 86.8%

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

      4. sub-neg 86.8%

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

      5. metadata-eval 86.8%

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

      6. +-commutative 86.8%

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

      7. sub-neg 86.8%

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

      8. log1p-define 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. neg-mul-1 85.9%

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

      3. distribute-neg-in 85.9%

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

      4. unsub-neg 85.9%

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

   8. Simplified 85.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -53 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+119} \lor \neg \left(z \leq 6.3 \cdot 10^{+140}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.25e+119) (not (<= z 6.3e+140)))
   (-
    (* z (* y (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
    t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e+119) || !(z <= 6.3e+140)) {
		tmp = (z * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.25d+119)) .or. (.not. (z <= 6.3d+140))) then
        tmp = (z * (y * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e+119) || !(z <= 6.3e+140)) {
		tmp = (z * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.25e+119) or not (z <= 6.3e+140):
		tmp = (z * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.25e+119) || !(z <= 6.3e+140))
		tmp = Float64(Float64(z * Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.25e+119) || ~((z <= 6.3e+140)))
		tmp = (z * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.25e+119], N[Not[LessEqual[z, 6.3e+140]], $MachinePrecision]], N[(N[(z * N[(y * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e119 or 6.29999999999999972e140 < z

   1. Initial program 75.2%

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

   3. Taylor expanded in z around inf 75.2%

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

      1. sub-neg 75.2%

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

      2. log1p-define 99.8%

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

      3. +-commutative 99.8%

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

      4. mul-1-neg 99.8%

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

      5. unsub-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.7%

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

      9. +-commutative 99.7%

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

      10. sub-neg 99.7%

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

      11. log1p-define 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf 39.7%

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

   7. Taylor expanded in y around 0 63.9%

      \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

   if -1.25e119 < z < 6.29999999999999972e140

   1. Initial program 98.2%

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

   3. Taylor expanded in x around 0 65.5%

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

      1. +-commutative 65.5%

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

      2. mul-1-neg 65.5%

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

      3. unsub-neg 65.5%

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

      4. sub-neg 65.5%

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

      5. metadata-eval 65.5%

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

      6. +-commutative 65.5%

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

      7. sub-neg 65.5%

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

      8. log1p-define 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in y around 0 65.0%

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

      1. +-commutative 65.0%

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

      2. neg-mul-1 65.0%

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

      3. distribute-neg-in 65.0%

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

      4. unsub-neg 65.0%

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

   8. Simplified 65.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+119} \lor \neg \left(z \leq 6.3 \cdot 10^{+140}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. sub-neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. fma-define 99.3%

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

   5. mul-1-neg 99.3%

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

   6. fma-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. sub-neg 99.3%

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

   9. metadata-eval 99.3%

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

   10. +-commutative 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 9: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+193}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.3e+193) (- (* (log y) (+ -1.0 x)) t) (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.3e+193) {
		tmp = (log(y) * (-1.0 + x)) - 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 (z <= 3.3e+193) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.3e+193:
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.3e+193)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.3e+193], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.3e193

   1. Initial program 94.2%

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

   3. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. fma-define 99.6%

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

      5. mul-1-neg 99.6%

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

      6. fma-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

      10. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around inf 86.4%

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

      1. sub-neg 86.4%

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

      2. metadata-eval 86.4%

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

      3. +-commutative 86.4%

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

      4. mul-1-neg 86.4%

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

      5. unsub-neg 86.4%

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

   8. Simplified 86.4%

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

   9. Taylor expanded in y around 0 93.6%

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

   if 3.3e193 < z

   1. Initial program 65.0%

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

   3. Taylor expanded in z around inf 65.1%

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

      1. sub-neg 65.1%

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

      2. log1p-define 99.9%

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

      3. +-commutative 99.9%

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

      4. mul-1-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/l* 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. log1p-define 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 47.5%

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

      1. sub-neg 47.5%

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

      2. log1p-undefine 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+193}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.0% 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 91.7%

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

3. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. sub-neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. fma-define 99.3%

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

   5. mul-1-neg 99.3%

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

   6. fma-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. sub-neg 99.3%

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

   9. metadata-eval 99.3%

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

   10. +-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in z around inf 99.1%

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

7. Final simplification 99.1%

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

Alternative 11: 42.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+25} \lor \neg \left(t \leq 4.5 \cdot 10^{-21}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.3e+25) (not (<= t 4.5e-21))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.3e+25) || !(t <= 4.5e-21)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.3d+25)) .or. (.not. (t <= 4.5d-21))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.3e+25) || !(t <= 4.5e-21)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.3e+25) or not (t <= 4.5e-21):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.3e+25) || !(t <= 4.5e-21))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.3e+25) || ~((t <= 4.5e-21)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.3e+25], N[Not[LessEqual[t, 4.5e-21]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2999999999999998e25 or 4.49999999999999968e-21 < t

   1. Initial program 96.2%

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

   3. Taylor expanded in x around 0 81.6%

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

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. sub-neg 81.6%

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

      5. metadata-eval 81.6%

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

      6. +-commutative 81.6%

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

      7. sub-neg 81.6%

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

      8. log1p-define 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in t around inf 80.2%

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

      1. neg-mul-1 80.2%

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

   8. Simplified 80.2%

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

   if -2.2999999999999998e25 < t < 4.49999999999999968e-21

   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. Add Preprocessing

   3. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. fma-define 99.3%

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

      5. mul-1-neg 99.3%

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

      6. fma-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

      10. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in z around inf 15.9%

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

      1. associate-*r* 15.9%

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

      2. neg-mul-1 15.9%

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

   8. Simplified 15.9%

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

   9. Taylor expanded in y around inf 15.7%

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

       1. associate-*r* 15.7%

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

       2. mul-1-neg 15.7%

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

   11. Simplified 15.7%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+25} \lor \neg \left(t \leq 4.5 \cdot 10^{-21}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.1% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. sub-neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. fma-define 99.3%

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

   5. mul-1-neg 99.3%

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

   6. fma-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. sub-neg 99.3%

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

   9. metadata-eval 99.3%

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

   10. +-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in z around inf 87.2%

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

   1. sub-neg 87.2%

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

   2. metadata-eval 87.2%

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

   3. +-commutative 87.2%

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

   4. mul-1-neg 87.2%

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

   5. unsub-neg 87.2%

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

8. Simplified 87.2%

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

9. Taylor expanded in x around inf 70.0%

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

    1. associate-/l* 69.6%

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

11. Simplified 69.6%

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

12. Taylor expanded in y around inf 50.4%

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

    1. sub-neg 50.4%

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

    2. metadata-eval 50.4%

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

    3. distribute-rgt-in 50.4%

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

    4. lft-mult-inverse 50.4%

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

    5. neg-mul-1 50.4%

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

    6. unsub-neg 50.4%

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

14. Simplified 50.4%

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

15. Final simplification 50.4%

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

Alternative 13: 45.9% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. sub-neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. fma-define 99.3%

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

   5. mul-1-neg 99.3%

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

   6. fma-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. sub-neg 99.3%

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

   9. metadata-eval 99.3%

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

   10. +-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in z around inf 50.2%

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

   1. associate-*r* 50.2%

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

   2. neg-mul-1 50.2%

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

8. Simplified 50.2%

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

9. Final simplification 50.2%

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

Alternative 14: 35.4% 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 91.7%

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

3. Taylor expanded in x around 0 59.7%

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

   1. +-commutative 59.7%

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

   2. mul-1-neg 59.7%

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

   3. unsub-neg 59.7%

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

   4. sub-neg 59.7%

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

   5. metadata-eval 59.7%

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

   6. +-commutative 59.7%

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

   7. sub-neg 59.7%

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

   8. log1p-define 67.6%

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

5. Simplified 67.6%

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

6. Taylor expanded in t around inf 41.9%

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

   1. neg-mul-1 41.9%

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

8. Simplified 41.9%

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

9. Final simplification 41.9%

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

Reproduce

herbie shell --seed 2024071 
(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))