Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Percentage Accurate: 85.0% → 99.8%

Time: 14.4s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. +-commutative 81.9%

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

   2. associate--l+ 81.9%

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

   3. fma-define 81.9%

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

   4. sub-neg 81.9%

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

   5. log1p-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in y around 0 99.6%

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

   1. associate-*r* 99.6%

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

   2. distribute-rgt-out 99.7%

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

   3. +-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. fma-define 99.7%

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

5. Simplified 99.7%

   \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \left(z \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right)}\right) - t \]
6. Add Preprocessing

Alternative 3: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-84} \lor \neg \left(x \leq 7.9 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6e-84) (not (<= x 7.9e-59)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6e-84) || !(x <= 7.9e-59)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6e-84) || !(x <= 7.9e-59)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6e-84) or not (x <= 7.9e-59):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6e-84) || !(x <= 7.9e-59))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6e-84], N[Not[LessEqual[x, 7.9e-59]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.0000000000000002e-84 or 7.89999999999999985e-59 < x

   1. Initial program 91.6%

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

      1. +-commutative 91.6%

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

      2. associate--l+ 91.6%

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

      3. fma-define 91.6%

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

      4. sub-neg 91.6%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

      1. neg-sub0 99.7%

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

      2. flip-- 97.8%

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

      3. metadata-eval 97.8%

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

      4. pow2 97.8%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - \color{blue}{{y}^{2}}}{0 + y}\right), x \cdot \log y - t\right) \]

      5. add-sqr-sqrt 97.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right), x \cdot \log y - t\right) \]

      6. sqrt-unprod 55.4%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{y \cdot y}}}\right), x \cdot \log y - t\right) \]

      7. sqr-neg 55.4%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}}\right), x \cdot \log y - t\right) \]

      8. sqrt-unprod 0.0%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}\right), x \cdot \log y - t\right) \]

      9. add-sqr-sqrt 90.6%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\left(-y\right)}}\right), x \cdot \log y - t\right) \]

      10. sub-neg 90.6%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{0 - y}}\right), x \cdot \log y - t\right) \]

      11. neg-sub0 90.6%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{-y}}\right), x \cdot \log y - t\right) \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}\right), x \cdot \log y - t\right) \]

      13. sqrt-unprod 55.4%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}\right), x \cdot \log y - t\right) \]

      14. sqr-neg 55.4%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\sqrt{\color{blue}{y \cdot y}}}\right), x \cdot \log y - t\right) \]

      15. sqrt-unprod 97.7%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right), x \cdot \log y - t\right) \]

      16. add-sqr-sqrt 97.8%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{y}}\right), x \cdot \log y - t\right) \]

   6. Applied egg-rr 97.8%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\frac{0 - {y}^{2}}{y}}\right), x \cdot \log y - t\right) \]
   7. Step-by-step derivation

      1. sub0-neg 97.8%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{\color{blue}{-{y}^{2}}}{y}\right), x \cdot \log y - t\right) \]

   8. Simplified 97.8%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\frac{-{y}^{2}}{y}}\right), x \cdot \log y - t\right) \]

   9. Taylor expanded in z around 0 90.7%

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

   if -6.0000000000000002e-84 < x < 7.89999999999999985e-59

   1. Initial program 67.9%

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

   3. Taylor expanded in x around 0 60.5%

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

      1. sub-neg 60.5%

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

      2. log1p-define 92.4%

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

   5. Simplified 92.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-84} \lor \neg \left(x \leq 7.9 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-83} \lor \neg \left(x \leq 1.9 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.85e-83) (not (<= x 1.9e-59)))
   (- (* x (log y)) t)
   (-
    (* y (* z (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.85e-83) || !(x <= 1.9e-59)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - 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 <= (-2.85d-83)) .or. (.not. (x <= 1.9d-59))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y * (z * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.85e-83) || !(x <= 1.9e-59)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.85e-83) or not (x <= 1.9e-59):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.85e-83) || !(x <= 1.9e-59))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.85e-83) || ~((x <= 1.9e-59)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.85e-83], N[Not[LessEqual[x, 1.9e-59]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85e-83 or 1.89999999999999992e-59 < x

   1. Initial program 91.6%

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

      1. +-commutative 91.6%

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

      2. associate--l+ 91.6%

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

      3. fma-define 91.6%

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

      4. sub-neg 91.6%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

      1. neg-sub0 99.7%

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

      2. flip-- 97.8%

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

      3. metadata-eval 97.8%

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

      4. pow2 97.8%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - \color{blue}{{y}^{2}}}{0 + y}\right), x \cdot \log y - t\right) \]

      5. add-sqr-sqrt 97.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right), x \cdot \log y - t\right) \]

      6. sqrt-unprod 55.4%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{y \cdot y}}}\right), x \cdot \log y - t\right) \]

      7. sqr-neg 55.4%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}}\right), x \cdot \log y - t\right) \]

      8. sqrt-unprod 0.0%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}\right), x \cdot \log y - t\right) \]

      9. add-sqr-sqrt 90.6%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\left(-y\right)}}\right), x \cdot \log y - t\right) \]

      10. sub-neg 90.6%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{0 - y}}\right), x \cdot \log y - t\right) \]

      11. neg-sub0 90.6%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{-y}}\right), x \cdot \log y - t\right) \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}\right), x \cdot \log y - t\right) \]

      13. sqrt-unprod 55.4%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}\right), x \cdot \log y - t\right) \]

      14. sqr-neg 55.4%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\sqrt{\color{blue}{y \cdot y}}}\right), x \cdot \log y - t\right) \]

      15. sqrt-unprod 97.7%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right), x \cdot \log y - t\right) \]

      16. add-sqr-sqrt 97.8%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{y}}\right), x \cdot \log y - t\right) \]

   6. Applied egg-rr 97.8%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\frac{0 - {y}^{2}}{y}}\right), x \cdot \log y - t\right) \]
   7. Step-by-step derivation

      1. sub0-neg 97.8%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{\color{blue}{-{y}^{2}}}{y}\right), x \cdot \log y - t\right) \]

   8. Simplified 97.8%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\frac{-{y}^{2}}{y}}\right), x \cdot \log y - t\right) \]

   9. Taylor expanded in z around 0 90.7%

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

   if -2.85e-83 < x < 1.89999999999999992e-59

   1. Initial program 67.9%

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

   3. Taylor expanded in x around 0 60.5%

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

      1. sub-neg 60.5%

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

      2. log1p-define 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in y around 0 92.4%

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

   7. Taylor expanded in z around 0 92.4%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-83} \lor \neg \left(x \leq 1.9 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+16} \lor \neg \left(x \leq 116\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e+16) (not (<= x 116.0)))
   (* x (log y))
   (-
    (* y (* z (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e+16) || !(x <= 116.0)) {
		tmp = x * log(y);
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - 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 <= (-2d+16)) .or. (.not. (x <= 116.0d0))) then
        tmp = x * log(y)
    else
        tmp = (y * (z * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e+16) || !(x <= 116.0)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2e+16) or not (x <= 116.0):
		tmp = x * math.log(y)
	else:
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e+16) || !(x <= 116.0))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2e+16) || ~((x <= 116.0)))
		tmp = x * log(y);
	else
		tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e+16], N[Not[LessEqual[x, 116.0]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e16 or 116 < x

   1. Initial program 95.4%

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

      1. +-commutative 95.4%

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

      2. associate--l+ 95.4%

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

      3. fma-define 95.4%

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

      4. sub-neg 95.4%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

      1. neg-sub0 99.7%

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

      2. flip-- 99.7%

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

      3. metadata-eval 99.7%

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

      4. pow2 99.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - \color{blue}{{y}^{2}}}{0 + y}\right), x \cdot \log y - t\right) \]

      5. add-sqr-sqrt 99.6%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right), x \cdot \log y - t\right) \]

      6. sqrt-unprod 55.6%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{y \cdot y}}}\right), x \cdot \log y - t\right) \]

      7. sqr-neg 55.6%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}}\right), x \cdot \log y - t\right) \]

      8. sqrt-unprod 0.0%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}\right), x \cdot \log y - t\right) \]

      9. add-sqr-sqrt 95.2%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{0 + \color{blue}{\left(-y\right)}}\right), x \cdot \log y - t\right) \]

      10. sub-neg 95.2%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{0 - y}}\right), x \cdot \log y - t\right) \]

      11. neg-sub0 95.2%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{-y}}\right), x \cdot \log y - t\right) \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}\right), x \cdot \log y - t\right) \]

      13. sqrt-unprod 55.6%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}\right), x \cdot \log y - t\right) \]

      14. sqr-neg 55.6%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\sqrt{\color{blue}{y \cdot y}}}\right), x \cdot \log y - t\right) \]

      15. sqrt-unprod 99.6%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right), x \cdot \log y - t\right) \]

      16. add-sqr-sqrt 99.7%

          \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{0 - {y}^{2}}{\color{blue}{y}}\right), x \cdot \log y - t\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\frac{0 - {y}^{2}}{y}}\right), x \cdot \log y - t\right) \]
   7. Step-by-step derivation

      1. sub0-neg 99.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\frac{\color{blue}{-{y}^{2}}}{y}\right), x \cdot \log y - t\right) \]

   8. Simplified 99.7%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\frac{-{y}^{2}}{y}}\right), x \cdot \log y - t\right) \]

   9. Taylor expanded in x around inf 73.3%

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

   if -2e16 < x < 116

   1. Initial program 71.6%

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

   3. Taylor expanded in x around 0 58.2%

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

      1. sub-neg 58.2%

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

      2. log1p-define 86.5%

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

   5. Simplified 86.5%

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

   6. Taylor expanded in y around 0 86.5%

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

   7. Taylor expanded in z around 0 86.5%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+16} \lor \neg \left(x \leq 116\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in y around 0 99.2%

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

   1. +-commutative 99.2%

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

   2. mul-1-neg 99.2%

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

   3. unsub-neg 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 7: 57.4% accurate, 11.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * (z * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in x around 0 42.9%

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

   1. sub-neg 42.9%

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

   2. log1p-define 60.9%

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

5. Simplified 60.9%

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

6. Taylor expanded in y around 0 60.9%

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

7. Taylor expanded in z around 0 60.9%

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

8. Final simplification 60.9%

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

Alternative 8: 57.4% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * (z * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in x around 0 42.9%

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

   1. sub-neg 42.9%

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

   2. log1p-define 60.9%

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

5. Simplified 60.9%

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

6. Taylor expanded in y around 0 60.9%

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

7. Taylor expanded in z around 0 60.9%

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

8. Taylor expanded in y around 0 60.8%

   \[\leadsto y \cdot \left(z \cdot \left(y \cdot \left(\color{blue}{-0.3333333333333333 \cdot y} - 0.5\right) - 1\right)\right) - t \]
9. Step-by-step derivation

   1. *-commutative 60.8%

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

10. Simplified 60.8%

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

11. Final simplification 60.8%

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

Alternative 9: 57.3% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in x around 0 42.9%

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

   1. sub-neg 42.9%

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

   2. log1p-define 60.9%

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

5. Simplified 60.9%

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

6. Taylor expanded in y around 0 60.9%

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

7. Taylor expanded in y around 0 60.7%

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

   1. associate-*r* 60.7%

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

   2. distribute-rgt-out 60.7%

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

   3. *-commutative 60.7%

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

9. Simplified 60.7%

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

Alternative 10: 57.0% accurate, 35.2× 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 81.9%

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

3. Taylor expanded in x around 0 42.9%

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

   1. sub-neg 42.9%

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

   2. log1p-define 60.9%

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

5. Simplified 60.9%

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

6. Taylor expanded in y around 0 60.3%

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

   1. associate-*r* 60.3%

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

   2. neg-mul-1 60.3%

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

8. Simplified 60.3%

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

9. Final simplification 60.3%

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

Alternative 11: 42.4% accurate, 105.5× 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 81.9%

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

3. Taylor expanded in x around 0 42.9%

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

   1. sub-neg 42.9%

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

   2. log1p-define 60.9%

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

5. Simplified 60.9%

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

6. Taylor expanded in z around 0 41.7%

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

   1. neg-mul-1 41.7%

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

8. Simplified 41.7%

   \[\leadsto \color{blue}{-t} \]
9. Add Preprocessing

Alternative 12: 2.2% accurate, 211.0× 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 t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in x around 0 42.9%

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

   1. sub-neg 42.9%

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

   2. log1p-define 60.9%

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

5. Simplified 60.9%

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

6. Taylor expanded in z around 0 41.7%

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

   1. neg-mul-1 41.7%

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

8. Simplified 41.7%

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

   1. add-sqr-sqrt 18.9%

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

   2. sqrt-unprod 11.1%

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

   3. sqr-neg 11.1%

      \[\leadsto \sqrt{\color{blue}{t \cdot t}} \]

   4. sqrt-unprod 1.2%

      \[\leadsto \color{blue}{\sqrt{t} \cdot \sqrt{t}} \]

   5. add-sqr-sqrt 2.5%

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

   6. *-un-lft-identity 2.5%

      \[\leadsto \color{blue}{1 \cdot t} \]

10. Applied egg-rr 2.5%

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

    1. *-lft-identity 2.5%

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

12. Simplified 2.5%

    \[\leadsto \color{blue}{t} \]
13. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[((-z) * N[(N[(N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(1.0 * N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024186 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))