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

Percentage Accurate: 88.8% → 99.8%

Time: 13.5s

Alternatives: 15

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: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

   1. +-commutative 88.3%

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

   2. associate--l+ 88.3%

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

   3. fma-def 88.3%

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

   4. sub-neg 88.3%

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

   5. log1p-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	return (((x + -1.0) * Math.log(y)) + (z * Math.log1p(-y))) - t;
}

Python

def code(x, y, z, t):
	return (((x + -1.0) * math.log(y)) + (z * math.log1p(-y))) - t

Julia

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

2. Taylor expanded in z around inf 88.0%

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

   1. *-commutative 88.0%

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

   2. sub-neg 88.0%

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

   3. mul-1-neg 88.0%

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

   4. log1p-def 99.6%

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

   5. mul-1-neg 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 88.3%

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

2. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

   2. unsub-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. unpow2 99.4%

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

   5. associate-*l* 99.4%

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

4. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 94.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -0.004:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x -1.0) -2e+53)
   (- (* x (log y)) t)
   (if (<= (+ x -1.0) -0.004)
     (- (- (- y (* z y)) (log y)) t)
     (- (* (+ x -1.0) (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + -1.0) <= -2e+53) {
		tmp = (x * log(y)) - t;
	} else if ((x + -1.0) <= -0.004) {
		tmp = ((y - (z * y)) - log(y)) - t;
	} else {
		tmp = ((x + -1.0) * log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x + (-1.0d0)) <= (-2d+53)) then
        tmp = (x * log(y)) - t
    else if ((x + (-1.0d0)) <= (-0.004d0)) then
        tmp = ((y - (z * y)) - log(y)) - t
    else
        tmp = ((x + (-1.0d0)) * log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + -1.0) <= -2e+53) {
		tmp = (x * Math.log(y)) - t;
	} else if ((x + -1.0) <= -0.004) {
		tmp = ((y - (z * y)) - Math.log(y)) - t;
	} else {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + -1.0) <= -2e+53:
		tmp = (x * math.log(y)) - t
	elif (x + -1.0) <= -0.004:
		tmp = ((y - (z * y)) - math.log(y)) - t
	else:
		tmp = ((x + -1.0) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -2e+53)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(x + -1.0) <= -0.004)
		tmp = Float64(Float64(Float64(y - Float64(z * y)) - log(y)) - t);
	else
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + -1.0) <= -2e+53)
		tmp = (x * log(y)) - t;
	elseif ((x + -1.0) <= -0.004)
		tmp = ((y - (z * y)) - log(y)) - t;
	else
		tmp = ((x + -1.0) * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + -1.0), $MachinePrecision], -2e+53], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x + -1.0), $MachinePrecision], -0.004], N[(N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -2e53

   1. Initial program 93.4%

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

   2. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-*r* 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

      8. +-commutative 99.4%

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

      9. distribute-lft-in 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. sub-neg 99.4%

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

      13. metadata-eval 99.4%

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

      14. +-commutative 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around inf 91.0%

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

   if -2e53 < (-.f64 x 1) < -0.0040000000000000001

   1. Initial program 85.4%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. mul-1-neg 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in z around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. mul-1-neg 98.1%

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

      3. unsub-neg 98.1%

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

   10. Simplified 98.1%

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

   if -0.0040000000000000001 < (-.f64 x 1)

   1. Initial program 89.9%

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

      1. +-commutative 89.9%

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

      2. associate--l+ 89.9%

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

      3. fma-def 89.9%

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

      4. sub-neg 89.9%

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

      5. log1p-def 99.8%

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

   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 y around 0 88.6%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+53}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -0.004:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

2. Taylor expanded in z around inf 88.0%

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

   1. *-commutative 88.0%

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

   2. sub-neg 88.0%

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

   3. mul-1-neg 88.0%

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

   4. log1p-def 99.6%

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

   5. mul-1-neg 99.6%

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

4. Simplified 99.6%

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

5. Taylor expanded in y around 0 99.1%

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

   1. unpow2 99.1%

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

   2. associate-*r* 99.1%

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

   3. associate-*r* 99.1%

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

   4. distribute-rgt-in 99.2%

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

   5. mul-1-neg 99.2%

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

   6. sub-neg 99.2%

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

   7. *-commutative 99.2%

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

   8. associate-*l* 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 6: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.98\right):\\ \;\;\;\;\left(x \cdot \log y - z \cdot y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.98)))
   (- (- (* x (log y)) (* z y)) t)
   (- (- (- y (* z y)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.98)) {
		tmp = ((x * log(y)) - (z * y)) - t;
	} else {
		tmp = ((y - (z * y)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.98d0))) then
        tmp = ((x * log(y)) - (z * y)) - t
    else
        tmp = ((y - (z * y)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.98)) {
		tmp = ((x * Math.log(y)) - (z * y)) - t;
	} else {
		tmp = ((y - (z * y)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.98):
		tmp = ((x * math.log(y)) - (z * y)) - t
	else:
		tmp = ((y - (z * y)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.98))
		tmp = Float64(Float64(Float64(x * log(y)) - Float64(z * y)) - t);
	else
		tmp = Float64(Float64(Float64(y - Float64(z * y)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.98)))
		tmp = ((x * log(y)) - (z * y)) - t;
	else
		tmp = ((y - (z * y)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.98]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.97999999999999998 < x

   1. Initial program 90.0%

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

   2. Taylor expanded in z around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. sub-neg 90.0%

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

      3. mul-1-neg 90.0%

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

      4. log1p-def 99.7%

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

      5. mul-1-neg 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. associate-*r* 98.6%

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

      2. mul-1-neg 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 0.97999999999999998

   1. Initial program 86.5%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in z around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

   10. Simplified 98.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.98\right):\\ \;\;\;\;\left(x \cdot \log y - z \cdot y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \]

Alternative 7: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

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

Derivation

1. Initial program 88.3%

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

2. Taylor expanded in y around 0 98.9%

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

   1. mul-1-neg 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 8: 88.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+185}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.1e+185)
   (- (* z (- y)) t)
   (if (<= z 4e+284) (- (* (+ x -1.0) (log y)) t) (- (* z (log1p (- y))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.1e+185) {
		tmp = (z * -y) - t;
	} else if (z <= 4e+284) {
		tmp = ((x + -1.0) * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.1e+185) {
		tmp = (z * -y) - t;
	} else if (z <= 4e+284) {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.1e+185:
		tmp = (z * -y) - t
	elif z <= 4e+284:
		tmp = ((x + -1.0) * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.1e+185)
		tmp = Float64(Float64(z * Float64(-y)) - t);
	elseif (z <= 4e+284)
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.1e+185], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 4e+284], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1e185

   1. Initial program 39.7%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 92.4%

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

      1. mul-1-neg 92.4%

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

   7. Simplified 92.4%

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

   8. Taylor expanded in z around inf 92.4%

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

      1. mul-1-neg 92.4%

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

      2. distribute-rgt-neg-in 92.4%

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

   10. Simplified 92.4%

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

   if -1.1e185 < z < 4.00000000000000032e284

   1. Initial program 93.1%

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

      1. +-commutative 93.1%

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

      2. associate--l+ 93.1%

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

      3. fma-def 93.1%

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

      4. sub-neg 93.1%

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

      5. log1p-def 99.8%

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

   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 y around 0 92.3%

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

   if 4.00000000000000032e284 < z

   1. Initial program 55.2%

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

   2. Taylor expanded in z around inf 55.2%

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

      1. *-commutative 55.2%

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

      2. sub-neg 55.2%

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

      3. mul-1-neg 55.2%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 90.7%

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

      1. mul-1-neg 74.2%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in z around inf 44.3%

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

      1. sub-neg 44.3%

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

      2. log1p-def 90.7%

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

   10. Simplified 90.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+185}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 9: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

2. Taylor expanded in z around inf 88.0%

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

   1. *-commutative 88.0%

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

   2. sub-neg 88.0%

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

   3. mul-1-neg 88.0%

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

   4. log1p-def 99.6%

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

   5. mul-1-neg 99.6%

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

4. Simplified 99.6%

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

5. Taylor expanded in y around 0 98.7%

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

   1. associate-*r* 98.7%

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

   2. mul-1-neg 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 10: 86.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00375 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.00375) (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 <= -0.00375) || !(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 <= (-0.00375d0)) .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 <= -0.00375) || !(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 <= -0.00375) 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 <= -0.00375) || !(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 <= -0.00375) || ~((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, -0.00375], 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 < -0.0037499999999999999 or 1 < x

   1. Initial program 89.9%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

      3. associate-*r* 99.2%

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

      4. *-commutative 99.2%

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

      5. unpow2 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. sub-neg 99.2%

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

      13. metadata-eval 99.2%

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

      14. +-commutative 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around inf 86.8%

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

   if -0.0037499999999999999 < x < 1

   1. Initial program 86.6%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around 0 84.9%

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

      1. mul-1-neg 84.9%

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

   10. Simplified 84.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00375 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]

Alternative 11: 87.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00375 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.00375) (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 <= -0.00375) || !(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 <= (-0.00375d0)) .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 <= -0.00375) || !(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 <= -0.00375) 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 <= -0.00375) || !(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 <= -0.00375) || ~((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, -0.00375], 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 < -0.0037499999999999999 or 1 < x

   1. Initial program 89.9%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

      3. associate-*r* 99.2%

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

      4. *-commutative 99.2%

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

      5. unpow2 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. sub-neg 99.2%

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

      13. metadata-eval 99.2%

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

      14. +-commutative 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around inf 86.8%

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

   if -0.0037499999999999999 < x < 1

   1. Initial program 86.6%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in z around 0 85.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00375 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]

Alternative 12: 60.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+171} \lor \neg \left(z \leq 8 \cdot 10^{+132}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.8e+171) (not (<= z 8e+132)))
   (- (* z (- y)) t)
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.8e+171) || !(z <= 8e+132)) {
		tmp = (z * -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 ((z <= (-1.8d+171)) .or. (.not. (z <= 8d+132))) then
        tmp = (z * -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 ((z <= -1.8e+171) || !(z <= 8e+132)) {
		tmp = (z * -y) - t;
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.8e+171) or not (z <= 8e+132):
		tmp = (z * -y) - t
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.8e+171) || !(z <= 8e+132))
		tmp = Float64(Float64(z * Float64(-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 ((z <= -1.8e+171) || ~((z <= 8e+132)))
		tmp = (z * -y) - t;
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.8e+171], N[Not[LessEqual[z, 8e+132]], $MachinePrecision]], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.80000000000000009e171 or 7.99999999999999993e132 < z

   1. Initial program 61.2%

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

   2. Taylor expanded in y around 0 97.2%

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

      1. mul-1-neg 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in x around 0 67.4%

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

      1. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in z around inf 64.0%

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

      1. mul-1-neg 64.0%

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

      2. distribute-rgt-neg-in 64.0%

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

   10. Simplified 64.0%

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

   if -1.80000000000000009e171 < z < 7.99999999999999993e132

   1. Initial program 97.1%

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

   2. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around 0 64.1%

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

      1. mul-1-neg 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in y around 0 61.1%

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

      1. mul-1-neg 61.1%

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

   10. Simplified 61.1%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+171} \lor \neg \left(z \leq 8 \cdot 10^{+132}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]

Alternative 13: 46.4% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

2. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

   2. unsub-neg 99.4%

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

   3. associate-*r* 99.4%

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

   4. *-commutative 99.4%

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

   5. unpow2 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. distribute-lft-in 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. sub-neg 99.4%

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

   13. metadata-eval 99.4%

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

   14. +-commutative 99.4%

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

4. Simplified 99.4%

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

5. Taylor expanded in y around inf 46.5%

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

6. Taylor expanded in y around 0 46.1%

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

7. Final simplification 46.1%

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

Alternative 14: 46.2% 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 88.3%

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

2. Taylor expanded in y around 0 98.9%

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

   1. mul-1-neg 98.9%

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

4. Simplified 98.9%

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

5. Taylor expanded in x around 0 64.9%

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

   1. mul-1-neg 64.9%

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

7. Simplified 64.9%

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

8. Taylor expanded in z around inf 45.9%

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

   1. mul-1-neg 45.9%

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

   2. distribute-rgt-neg-in 45.9%

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

10. Simplified 45.9%

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

11. Final simplification 45.9%

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

Alternative 15: 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 88.3%

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

   1. +-commutative 88.3%

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

   2. associate--l+ 88.3%

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

   3. fma-def 88.3%

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

   4. sub-neg 88.3%

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

   5. log1p-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around inf 34.6%

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

   1. neg-mul-1 34.6%

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

6. Simplified 34.6%

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

7. Final simplification 34.6%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023187 
(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))