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

Percentage Accurate: 89.3% → 99.8%

Time: 22.2s

Alternatives: 16

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

   1. sub-neg 90.7%

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

   2. +-commutative 90.7%

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

   3. associate-+l+ 90.7%

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

   4. fma-def 90.7%

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

   5. sub-neg 90.7%

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

   6. metadata-eval 90.7%

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

   7. sub-neg 90.7%

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

   8. log1p-def 99.8%

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

   9. sub-neg 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log y \cdot \left(-1 + x\right) + -0.5 \cdot \left(\left(z + -1\right) \cdot {y}^{2}\right)\right) + y \cdot \left(1 - z\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (+ (* (log y) (+ -1.0 x)) (* -0.5 (* (+ z -1.0) (pow y 2.0))))
   (* y (- 1.0 z)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((log(y) * (-1.0 + x)) + (-0.5 * ((z + -1.0) * pow(y, 2.0)))) + (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 = (((log(y) * ((-1.0d0) + x)) + ((-0.5d0) * ((z + (-1.0d0)) * (y ** 2.0d0)))) + (y * (1.0d0 - z))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.log(y) * (-1.0 + x)) + (-0.5 * ((z + -1.0) * Math.pow(y, 2.0)))) + (y * (1.0 - z))) - t;
}

Python

def code(x, y, z, t):
	return (((math.log(y) * (-1.0 + x)) + (-0.5 * ((z + -1.0) * math.pow(y, 2.0)))) + (y * (1.0 - z))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(log(y) * Float64(-1.0 + x)) + Float64(-0.5 * Float64(Float64(z + -1.0) * (y ^ 2.0)))) + Float64(y * Float64(1.0 - z))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((log(y) * (-1.0 + x)) + (-0.5 * ((z + -1.0) * (y ^ 2.0)))) + (y * (1.0 - z))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(z + -1.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in y around 0 99.4%

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

3. Final simplification 99.4%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in y around 0 99.3%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
3. 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. *-commutative 99.3%

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

   5. mul-1-neg 99.3%

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

   6. unsub-neg 99.3%

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

   7. *-commutative 99.3%

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

   8. +-commutative 99.3%

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

   9. sub-neg 99.3%

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

   10. metadata-eval 99.3%

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

   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%

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

   1. fma-def 99.3%

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

   2. sub-neg 99.3%

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

   3. metadata-eval 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 4: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000000000000 \lor \neg \left(-1 + x \leq -0.99999999999\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - \left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -5000000000000.0) (not (<= (+ -1.0 x) -0.99999999999)))
   (- (* (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) <= -5000000000000.0) || !((-1.0 + x) <= -0.99999999999)) {
		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) <= (-5000000000000.0d0)) .or. (.not. (((-1.0d0) + x) <= (-0.99999999999d0)))) 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) <= -5000000000000.0) || !((-1.0 + x) <= -0.99999999999)) {
		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) <= -5000000000000.0) or not ((-1.0 + x) <= -0.99999999999):
		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) <= -5000000000000.0) || !(Float64(-1.0 + x) <= -0.99999999999))
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(y * Float64(1.0 - z)) - Float64(log(y) + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -5000000000000.0) || ~(((-1.0 + x) <= -0.99999999999)))
		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], -5000000000000.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.99999999999]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -5e12 or -0.99999999999 < (-.f64 x 1)

   1. Initial program 96.5%

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

   2. Taylor expanded in y around 0 96.5%

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

   if -5e12 < (-.f64 x 1) < -0.99999999999

   1. Initial program 84.9%

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

      1. sub-neg 84.9%

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

      2. +-commutative 84.9%

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

      3. associate-+l+ 84.9%

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

      4. fma-def 84.9%

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

      5. sub-neg 84.9%

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

      6. metadata-eval 84.9%

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

      7. sub-neg 84.9%

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

      8. log1p-def 100.0%

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

      9. sub-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. mul-1-neg 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. mul-1-neg 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

      5. sub-neg 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. +-commutative 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000000000000 \lor \neg \left(-1 + x \leq -0.99999999999\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - \left(\log y + t\right)\\ \end{array} \]

Alternative 5: 96.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(1 - z\right)\\ \mathbf{if}\;-1 + x \leq -5000000000000:\\ \;\;\;\;\left(x \cdot \log y + t_1\right) - t\\ \mathbf{elif}\;-1 + x \leq -0.99999999999:\\ \;\;\;\;t_1 - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- 1.0 z))))
   (if (<= (+ -1.0 x) -5000000000000.0)
     (- (+ (* x (log y)) t_1) t)
     (if (<= (+ -1.0 x) -0.99999999999)
       (- t_1 (+ (log y) t))
       (- (* (log y) (+ -1.0 x)) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (1.0 - z);
	double tmp;
	if ((-1.0 + x) <= -5000000000000.0) {
		tmp = ((x * log(y)) + t_1) - t;
	} else if ((-1.0 + x) <= -0.99999999999) {
		tmp = t_1 - (log(y) + t);
	} else {
		tmp = (log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (1.0d0 - z)
    if (((-1.0d0) + x) <= (-5000000000000.0d0)) then
        tmp = ((x * log(y)) + t_1) - t
    else if (((-1.0d0) + x) <= (-0.99999999999d0)) then
        tmp = t_1 - (log(y) + t)
    else
        tmp = (log(y) * ((-1.0d0) + x)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (1.0 - z);
	double tmp;
	if ((-1.0 + x) <= -5000000000000.0) {
		tmp = ((x * Math.log(y)) + t_1) - t;
	} else if ((-1.0 + x) <= -0.99999999999) {
		tmp = t_1 - (Math.log(y) + t);
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (1.0 - z)
	tmp = 0
	if (-1.0 + x) <= -5000000000000.0:
		tmp = ((x * math.log(y)) + t_1) - t
	elif (-1.0 + x) <= -0.99999999999:
		tmp = t_1 - (math.log(y) + t)
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(1.0 - z))
	tmp = 0.0
	if (Float64(-1.0 + x) <= -5000000000000.0)
		tmp = Float64(Float64(Float64(x * log(y)) + t_1) - t);
	elseif (Float64(-1.0 + x) <= -0.99999999999)
		tmp = Float64(t_1 - Float64(log(y) + t));
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (1.0 - z);
	tmp = 0.0;
	if ((-1.0 + x) <= -5000000000000.0)
		tmp = ((x * log(y)) + t_1) - t;
	elseif ((-1.0 + x) <= -0.99999999999)
		tmp = t_1 - (log(y) + t);
	else
		tmp = (log(y) * (-1.0 + x)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -5000000000000.0], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.99999999999], N[(t$95$1 - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -5e12

   1. Initial program 95.6%

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

   2. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. mul-1-neg 99.6%

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

      6. unsub-neg 99.6%

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

      7. *-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. sub-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. +-commutative 99.6%

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

   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 x around inf 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   if -5e12 < (-.f64 x 1) < -0.99999999999

   1. Initial program 84.9%

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

      1. sub-neg 84.9%

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

      2. +-commutative 84.9%

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

      3. associate-+l+ 84.9%

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

      4. fma-def 84.9%

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

      5. sub-neg 84.9%

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

      6. metadata-eval 84.9%

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

      7. sub-neg 84.9%

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

      8. log1p-def 100.0%

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

      9. sub-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. mul-1-neg 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. mul-1-neg 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

      5. sub-neg 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. +-commutative 98.4%

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

   9. Simplified 98.4%

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

   if -0.99999999999 < (-.f64 x 1)

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0 97.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5000000000000:\\ \;\;\;\;\left(x \cdot \log y + y \cdot \left(1 - z\right)\right) - t\\ \mathbf{elif}\;-1 + x \leq -0.99999999999:\\ \;\;\;\;y \cdot \left(1 - z\right) - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]

Alternative 6: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in y around 0 99.3%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
3. 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. *-commutative 99.3%

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

   5. mul-1-neg 99.3%

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

   6. unsub-neg 99.3%

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

   7. *-commutative 99.3%

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

   8. +-commutative 99.3%

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

   9. sub-neg 99.3%

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

   10. metadata-eval 99.3%

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

   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. Final simplification 99.3%

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

Alternative 7: 88.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq 10^{+178}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) 1e+178) (- (* (log y) (+ -1.0 x)) t) (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= 1e+178) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= 1e+178)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], 1e+178], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z 1) < 1.0000000000000001e178

   1. Initial program 93.6%

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

   2. Taylor expanded in y around 0 92.6%

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

   if 1.0000000000000001e178 < (-.f64 z 1)

   1. Initial program 62.9%

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

   2. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. mul-1-neg 99.8%

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

      6. unsub-neg 99.8%

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

      7. *-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. sub-neg 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in z around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-rgt-neg-in 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in y around 0 86.5%

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

      1. neg-mul-1 86.5%

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

      2. +-commutative 86.5%

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

      3. mul-1-neg 86.5%

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

      4. distribute-neg-in 86.5%

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

      5. fma-udef 86.6%

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

   10. Simplified 86.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 10^{+178}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 8: 87.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.07 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.07) (not (<= x 1.0))) (- (* x (log y)) t) (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.07) || !(x <= 1.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-0.07d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.07) || !(x <= 1.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -0.07) or not (x <= 1.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.07) || !(x <= 1.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -0.07) || ~((x <= 1.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.07], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.070000000000000007 or 1 < x

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 95.0%

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

   3. Taylor expanded in x around inf 92.1%

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

   if -0.070000000000000007 < x < 1

   1. Initial program 86.4%

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

      1. sub-neg 86.4%

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

      2. +-commutative 86.4%

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

      3. associate-+l+ 86.4%

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

      4. fma-def 86.4%

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

      5. sub-neg 86.4%

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

      6. metadata-eval 86.4%

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

      7. sub-neg 86.4%

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

      8. log1p-def 100.0%

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

      9. sub-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in y around 0 83.2%

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

      1. mul-1-neg 83.2%

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

      2. distribute-neg-in 83.2%

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

      3. +-commutative 83.2%

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

      4. sub-neg 83.2%

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

   9. Simplified 83.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.07 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]

Alternative 9: 87.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.07 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.07) (not (<= x 1.0)))
   (- (* x (log y)) t)
   (- (- y (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.07) || !(x <= 1.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-0.07d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.07) || !(x <= 1.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -0.07) or not (x <= 1.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.07) || !(x <= 1.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -0.07) || ~((x <= 1.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.07], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.070000000000000007 or 1 < x

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 95.0%

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

   3. Taylor expanded in x around inf 92.1%

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

   if -0.070000000000000007 < x < 1

   1. Initial program 86.4%

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

   2. Taylor expanded in y around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-commutative 98.9%

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

      5. mul-1-neg 98.9%

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

      6. unsub-neg 98.9%

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

      7. *-commutative 98.9%

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

      8. +-commutative 98.9%

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

      9. sub-neg 98.9%

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

      10. metadata-eval 98.9%

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

      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 x around 0 97.4%

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

      1. sub-neg 97.4%

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

      2. mul-1-neg 97.4%

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

      3. distribute-lft-in 97.4%

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

      4. mul-1-neg 97.4%

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

      5. +-commutative 97.4%

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

      6. fma-def 97.4%

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

      7. sub-neg 97.4%

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

      8. metadata-eval 97.4%

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

      9. +-commutative 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in z around 0 83.4%

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

      1. neg-mul-1 83.4%

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

      2. unsub-neg 83.4%

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

   10. Simplified 83.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.07 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]

Alternative 10: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+166}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+177}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.6e+166)
   (- (* z (- y)) t)
   (if (<= z 8.2e+177) (- (- t) (log y)) (- (fma y z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e+166) {
		tmp = (z * -y) - t;
	} else if (z <= 8.2e+177) {
		tmp = -t - log(y);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.6e+166)
		tmp = Float64(Float64(z * Float64(-y)) - t);
	elseif (z <= 8.2e+177)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.6e+166], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 8.2e+177], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5999999999999999e166

   1. Initial program 73.5%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. mul-1-neg 99.9%

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

      6. unsub-neg 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around inf 59.7%

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

      1. mul-1-neg 59.7%

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

      2. distribute-rgt-neg-in 59.7%

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

   7. Simplified 59.7%

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

   if -2.5999999999999999e166 < z < 8.20000000000000029e177

   1. Initial program 96.8%

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

      1. sub-neg 96.8%

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

      2. +-commutative 96.8%

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

      3. associate-+l+ 96.8%

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

      4. fma-def 96.8%

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

      5. sub-neg 96.8%

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

      6. metadata-eval 96.8%

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

      7. sub-neg 96.8%

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

      8. log1p-def 99.8%

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

      9. sub-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 59.5%

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

      1. mul-1-neg 59.5%

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

   6. Simplified 59.5%

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

   7. Taylor expanded in y around 0 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-neg-in 55.6%

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

      3. +-commutative 55.6%

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

      4. sub-neg 55.6%

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

   9. Simplified 55.6%

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

   if 8.20000000000000029e177 < z

   1. Initial program 62.9%

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

   2. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. mul-1-neg 99.8%

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

      6. unsub-neg 99.8%

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

      7. *-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. sub-neg 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in z around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. distribute-rgt-neg-in 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in y around 0 86.5%

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

      1. neg-mul-1 86.5%

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

      2. +-commutative 86.5%

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

      3. mul-1-neg 86.5%

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

      4. distribute-neg-in 86.5%

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

      5. fma-udef 86.6%

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

   10. Simplified 86.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+166}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+177}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 11: 42.7% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-19} \lor \neg \left(t \leq 4.9 \cdot 10^{+29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.15e-19) (not (<= t 4.9e+29))) (- t) (* y (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.15e-19) || !(t <= 4.9e+29)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.15d-19)) .or. (.not. (t <= 4.9d+29))) then
        tmp = -t
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.15e-19) || !(t <= 4.9e+29)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.15e-19) or not (t <= 4.9e+29):
		tmp = -t
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.15e-19) || !(t <= 4.9e+29))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.15e-19) || ~((t <= 4.9e+29)))
		tmp = -t;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.15e-19], N[Not[LessEqual[t, 4.9e+29]], $MachinePrecision]], (-t), N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.15e-19 or 4.9000000000000001e29 < t

   1. Initial program 97.8%

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

   2. Taylor expanded in t around inf 68.4%

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

      1. neg-mul-1 68.4%

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

   4. Simplified 68.4%

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

   if -2.15e-19 < t < 4.9000000000000001e29

   1. Initial program 84.1%

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

   2. Taylor expanded in y around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. metadata-eval 98.7%

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

      4. *-commutative 98.7%

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

      5. mul-1-neg 98.7%

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

      6. unsub-neg 98.7%

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

      7. *-commutative 98.7%

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

      8. +-commutative 98.7%

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

      9. sub-neg 98.7%

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

      10. metadata-eval 98.7%

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

      11. +-commutative 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in y around 0 98.7%

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

      1. fma-def 98.7%

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

      2. sub-neg 98.7%

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

      3. metadata-eval 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around inf 17.6%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-19} \lor \neg \left(t \leq 4.9 \cdot 10^{+29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 12: 42.7% accurate, 23.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.5e-19) (- (* z y) t) (if (<= t 4.7e+29) (* y (- 1.0 z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.5e-19) {
		tmp = (z * y) - t;
	} else if (t <= 4.7e+29) {
		tmp = y * (1.0 - z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.5d-19)) then
        tmp = (z * y) - t
    else if (t <= 4.7d+29) then
        tmp = y * (1.0d0 - z)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.5e-19) {
		tmp = (z * y) - t;
	} else if (t <= 4.7e+29) {
		tmp = y * (1.0 - z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.5e-19:
		tmp = (z * y) - t
	elif t <= 4.7e+29:
		tmp = y * (1.0 - z)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.5e-19)
		tmp = Float64(Float64(z * y) - t);
	elseif (t <= 4.7e+29)
		tmp = Float64(y * Float64(1.0 - z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.5e-19)
		tmp = (z * y) - t;
	elseif (t <= 4.7e+29)
		tmp = y * (1.0 - z);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.5e-19], N[(N[(z * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 4.7e+29], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < -2.5000000000000002e-19

   1. Initial program 98.3%

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

   2. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. mul-1-neg 99.8%

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

      6. unsub-neg 99.8%

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

      7. *-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. sub-neg 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in z around inf 70.4%

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

      1. mul-1-neg 70.4%

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

      2. distribute-rgt-neg-in 70.4%

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

   7. Simplified 70.4%

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

      1. sub-neg 70.4%

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

      2. add-sqr-sqrt 41.9%

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

      3. add-sqr-sqrt 70.4%

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

      4. add-sqr-sqrt 34.4%

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

      5. sqrt-unprod 54.1%

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

      6. sqr-neg 54.1%

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

      7. sqrt-unprod 34.6%

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

      8. add-sqr-sqrt 69.0%

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

   9. Applied egg-rr 69.0%

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

       1. sub-neg 69.0%

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

   11. Simplified 69.0%

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

   if -2.5000000000000002e-19 < t < 4.7000000000000002e29

   1. Initial program 84.1%

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

   2. Taylor expanded in y around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. metadata-eval 98.7%

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

      4. *-commutative 98.7%

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

      5. mul-1-neg 98.7%

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

      6. unsub-neg 98.7%

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

      7. *-commutative 98.7%

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

      8. +-commutative 98.7%

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

      9. sub-neg 98.7%

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

      10. metadata-eval 98.7%

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

      11. +-commutative 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in y around 0 98.7%

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

      1. fma-def 98.7%

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

      2. sub-neg 98.7%

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

      3. metadata-eval 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around inf 17.6%

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

   if 4.7000000000000002e29 < t

   1. Initial program 97.2%

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

   2. Taylor expanded in t around inf 67.6%

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

      1. neg-mul-1 67.6%

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

   4. Simplified 67.6%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 13: 42.4% accurate, 26.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-19} \lor \neg \left(t \leq 4.7 \cdot 10^{+29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.7e-19) (not (<= t 4.7e+29))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.7e-19) || !(t <= 4.7e+29)) {
		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 <= (-1.7d-19)) .or. (.not. (t <= 4.7d+29))) 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 <= -1.7e-19) || !(t <= 4.7e+29)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.7e-19) or not (t <= 4.7e+29):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.7e-19) || !(t <= 4.7e+29))
		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 <= -1.7e-19) || ~((t <= 4.7e+29)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.7e-19], N[Not[LessEqual[t, 4.7e+29]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7000000000000001e-19 or 4.7000000000000002e29 < t

   1. Initial program 97.8%

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

   2. Taylor expanded in t around inf 68.4%

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

      1. neg-mul-1 68.4%

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

   4. Simplified 68.4%

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

   if -1.7000000000000001e-19 < t < 4.7000000000000002e29

   1. Initial program 84.1%

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

   2. Taylor expanded in y around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. metadata-eval 98.7%

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

      4. *-commutative 98.7%

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

      5. mul-1-neg 98.7%

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

      6. unsub-neg 98.7%

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

      7. *-commutative 98.7%

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

      8. +-commutative 98.7%

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

      9. sub-neg 98.7%

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

      10. metadata-eval 98.7%

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

      11. +-commutative 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in y around 0 98.7%

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

      1. fma-def 98.7%

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

      2. sub-neg 98.7%

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

      3. metadata-eval 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in z around inf 17.0%

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

      1. associate-*r* 17.0%

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

      2. neg-mul-1 17.0%

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

   10. Simplified 17.0%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-19} \lor \neg \left(t \leq 4.7 \cdot 10^{+29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 14: 45.9% 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 90.7%

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

2. Taylor expanded in y around 0 99.3%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
3. 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. *-commutative 99.3%

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

   5. mul-1-neg 99.3%

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

   6. unsub-neg 99.3%

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

   7. *-commutative 99.3%

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

   8. +-commutative 99.3%

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

   9. sub-neg 99.3%

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

   10. metadata-eval 99.3%

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

   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 inf 43.2%

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

6. Final simplification 43.2%

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

Alternative 15: 45.7% 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 90.7%

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

2. Taylor expanded in y around 0 99.3%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
3. 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. *-commutative 99.3%

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

   5. mul-1-neg 99.3%

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

   6. unsub-neg 99.3%

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

   7. *-commutative 99.3%

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

   8. +-commutative 99.3%

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

   9. sub-neg 99.3%

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

   10. metadata-eval 99.3%

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

   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 z around inf 43.1%

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

   1. mul-1-neg 43.1%

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

   2. distribute-rgt-neg-in 43.1%

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

7. Simplified 43.1%

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

8. Final simplification 43.1%

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

Alternative 16: 35.5% 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 90.7%

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

2. Taylor expanded in t around inf 34.7%

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

   1. neg-mul-1 34.7%

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

4. Simplified 34.7%

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

5. Final simplification 34.7%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023320 
(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))