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

Percentage Accurate: 85.0% → 99.8%

Time: 14.3s

Alternatives: 14

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 82.8%

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

   1. +-commutative 82.8%

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

   2. associate--l+ 82.8%

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

   3. fma-def 82.8%

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

   4. sub-neg 82.8%

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

   5. log1p-def 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. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

2. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. mul-1-neg 99.6%

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

   3. unsub-neg 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

2. Taylor expanded in y around 0 99.2%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)} - t \]
3. 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. fma-def 99.2%

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

   3. mul-1-neg 99.2%

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

   4. distribute-rgt-neg-in 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

2. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. mul-1-neg 99.6%

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

   3. unsub-neg 99.6%

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

4. Simplified 99.6%

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

5. Taylor expanded in y around 0 99.2%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)} - t \]
6. 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. fma-def 99.2%

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

   3. mul-1-neg 99.2%

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

   4. *-commutative 99.2%

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

   5. distribute-rgt-neg-in 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in x around 0 99.2%

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

   1. associate-*r* 99.2%

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

   2. log-pow 43.7%

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

   3. fma-def 43.7%

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

   4. neg-mul-1 43.7%

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

   5. log-pow 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 5: 89.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -580000 \lor \neg \left(t \leq 4.1 \cdot 10^{-97}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -580000.0) (not (<= t 4.1e-97))) (- t_1 t) (- t_1 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((t <= -580000.0) || !(t <= 4.1e-97)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if ((t <= (-580000.0d0)) .or. (.not. (t <= 4.1d-97))) then
        tmp = t_1 - t
    else
        tmp = t_1 - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if ((t <= -580000.0) || !(t <= 4.1e-97)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (t <= -580000.0) or not (t <= 4.1e-97):
		tmp = t_1 - t
	else:
		tmp = t_1 - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((t <= -580000.0) || !(t <= 4.1e-97))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((t <= -580000.0) || ~((t <= 4.1e-97)))
		tmp = t_1 - t;
	else
		tmp = t_1 - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -580000.0], N[Not[LessEqual[t, 4.1e-97]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.8e5 or 4.09999999999999993e-97 < t

   1. Initial program 94.9%

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

   2. Taylor expanded in y around 0 94.7%

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

   if -5.8e5 < t < 4.09999999999999993e-97

   1. Initial program 69.9%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. fma-def 98.8%

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

      3. mul-1-neg 98.8%

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

      4. *-commutative 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

      2. log-pow 36.7%

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

      3. fma-def 36.7%

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

      4. neg-mul-1 36.7%

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

      5. log-pow 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in t around 0 91.6%

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

       1. +-commutative 91.6%

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

       2. mul-1-neg 91.6%

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

       3. sub-neg 91.6%

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

   13. Simplified 91.6%

       \[\leadsto \color{blue}{x \cdot \log y - y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -580000 \lor \neg \left(t \leq 4.1 \cdot 10^{-97}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \end{array} \]

Alternative 6: 89.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-22} \lor \neg \left(x \leq 1.85 \cdot 10^{-131}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.2e-22) (not (<= x 1.85e-131)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.2e-22) || !(x <= 1.85e-131)) {
		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 <= -1.2e-22) || !(x <= 1.85e-131)) {
		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 <= -1.2e-22) or not (x <= 1.85e-131):
		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 <= -1.2e-22) || !(x <= 1.85e-131))
		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, -1.2e-22], N[Not[LessEqual[x, 1.85e-131]], $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 < -1.20000000000000001e-22 or 1.8500000000000001e-131 < x

   1. Initial program 90.1%

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

   2. Taylor expanded in y around 0 89.8%

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

   if -1.20000000000000001e-22 < x < 1.8500000000000001e-131

   1. Initial program 69.3%

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

   2. Taylor expanded in x around 0 58.9%

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

      1. sub-neg 58.9%

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

      2. mul-1-neg 58.9%

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

      3. log1p-def 89.4%

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

      4. mul-1-neg 89.4%

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

   4. Simplified 89.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-22} \lor \neg \left(x \leq 1.85 \cdot 10^{-131}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 7: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-93} \lor \neg \left(x \leq 2 \cdot 10^{-127}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e-93) (not (<= x 2e-127)))
   (- (* x (log y)) t)
   (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e-93) || !(x <= 2e-127)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * -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 <= (-2d-93)) .or. (.not. (x <= 2d-127))) then
        tmp = (x * log(y)) - t
    else
        tmp = (z * -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 <= -2e-93) || !(x <= 2e-127)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2e-93) or not (x <= 2e-127):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e-93) || !(x <= 2e-127))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(-y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2e-93) || ~((x <= 2e-127)))
		tmp = (x * log(y)) - t;
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e-93], N[Not[LessEqual[x, 2e-127]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999998e-93 or 2.0000000000000001e-127 < x

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0 88.5%

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

   if -1.9999999999999998e-93 < x < 2.0000000000000001e-127

   1. Initial program 66.7%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around 0 91.3%

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

      1. mul-1-neg 91.3%

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

      2. *-commutative 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

   7. Simplified 91.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-93} \lor \neg \left(x \leq 2 \cdot 10^{-127}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]

Alternative 8: 99.1% 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 82.8%

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

2. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. mul-1-neg 99.6%

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

   3. unsub-neg 99.6%

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

4. Simplified 99.6%

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

5. Taylor expanded in y around 0 99.2%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)} - t \]
6. 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. sub-neg 99.2%

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

   4. *-commutative 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 9: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+107} \lor \neg \left(x \leq 1.8 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.2e+107) (not (<= x 1.8e+22))) (* x (log y)) (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.2e+107) || !(x <= 1.8e+22)) {
		tmp = x * log(y);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.2e+107) || !(x <= 1.8e+22))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.2e+107], N[Not[LessEqual[x, 1.8e+22]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000029e107 or 1.8e22 < x

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. mul-1-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 99.5%

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

      1. associate-*r* 99.5%

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

      2. log-pow 8.9%

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

      3. fma-def 8.9%

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

      4. neg-mul-1 8.9%

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

      5. log-pow 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in x around inf 80.4%

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

   if -3.20000000000000029e107 < x < 1.8e22

   1. Initial program 72.9%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-def 99.1%

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

      3. mul-1-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in x around 0 79.3%

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

      1. sub-neg 79.3%

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

      2. mul-1-neg 79.3%

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

      3. distribute-neg-in 79.3%

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

      4. fma-def 79.3%

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

   10. Simplified 79.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+107} \lor \neg \left(x \leq 1.8 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 10: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+107} \lor \neg \left(x \leq 2.5 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.1e+107) (not (<= x 2.5e+22)))
   (* x (log y))
   (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e+107) || !(x <= 2.5e+22)) {
		tmp = x * log(y);
	} else {
		tmp = (z * -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 <= (-3.1d+107)) .or. (.not. (x <= 2.5d+22))) then
        tmp = x * log(y)
    else
        tmp = (z * -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 <= -3.1e+107) || !(x <= 2.5e+22)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.1e+107) or not (x <= 2.5e+22):
		tmp = x * math.log(y)
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.1e+107) || !(x <= 2.5e+22))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(z * Float64(-y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.1e+107) || ~((x <= 2.5e+22)))
		tmp = x * log(y);
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.1e+107], N[Not[LessEqual[x, 2.5e+22]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000026e107 or 2.4999999999999998e22 < x

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. mul-1-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 99.5%

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

      1. associate-*r* 99.5%

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

      2. log-pow 8.9%

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

      3. fma-def 8.9%

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

      4. neg-mul-1 8.9%

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

      5. log-pow 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in x around inf 80.4%

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

   if -3.10000000000000026e107 < x < 2.4999999999999998e22

   1. Initial program 72.9%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. *-commutative 79.3%

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

      3. distribute-rgt-neg-in 79.3%

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

   7. Simplified 79.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+107} \lor \neg \left(x \leq 2.5 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]

Alternative 11: 48.8% accurate, 25.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-34} \lor \neg \left(t \leq 2.25 \cdot 10^{-85}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.06e-34) (not (<= t 2.25e-85))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.06e-34) || !(t <= 2.25e-85)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.06d-34)) .or. (.not. (t <= 2.25d-85))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.06e-34) || !(t <= 2.25e-85)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.06e-34) or not (t <= 2.25e-85):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.06e-34) || !(t <= 2.25e-85))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.06e-34) || ~((t <= 2.25e-85)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.06e-34], N[Not[LessEqual[t, 2.25e-85]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.06000000000000006e-34 or 2.25000000000000002e-85 < t

   1. Initial program 95.0%

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

   2. Taylor expanded in t around inf 60.1%

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

      1. neg-mul-1 60.1%

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

   4. Simplified 60.1%

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

   if -1.06000000000000006e-34 < t < 2.25000000000000002e-85

   1. Initial program 68.9%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. fma-def 98.8%

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

      3. mul-1-neg 98.8%

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

      4. *-commutative 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 98.7%

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

      1. associate-*r* 98.7%

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

      2. log-pow 37.0%

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

      3. fma-def 37.0%

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

      4. neg-mul-1 37.0%

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

      5. log-pow 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in y around inf 32.0%

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

       1. mul-1-neg 32.0%

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

       2. distribute-rgt-neg-in 32.0%

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

   13. Simplified 32.0%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-34} \lor \neg \left(t \leq 2.25 \cdot 10^{-85}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 12: 57.6% 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 82.8%

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

2. Taylor expanded in y around 0 99.2%

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

   1. mul-1-neg 99.2%

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

4. Simplified 99.2%

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

5. Taylor expanded in x around 0 52.9%

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

   1. mul-1-neg 52.9%

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

   2. *-commutative 52.9%

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

   3. distribute-rgt-neg-in 52.9%

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

7. Simplified 52.9%

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

8. Final simplification 52.9%

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

Alternative 13: 42.9% 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 82.8%

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

2. Taylor expanded in t around inf 36.2%

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

   1. neg-mul-1 36.2%

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

4. Simplified 36.2%

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

5. Final simplification 36.2%

   \[\leadsto -t \]

Alternative 14: 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 82.8%

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

2. Taylor expanded in y around 0 99.2%

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

   1. mul-1-neg 99.2%

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

4. Simplified 99.2%

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

5. Taylor expanded in x around 0 52.9%

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

   1. mul-1-neg 52.9%

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

   2. *-commutative 52.9%

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

   3. distribute-rgt-neg-in 52.9%

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

7. Simplified 52.9%

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

   1. sub-neg 52.9%

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

   2. *-commutative 52.9%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 36.0%

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

   5. sqr-neg 36.0%

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

   6. sqrt-unprod 35.9%

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

   7. add-sqr-sqrt 35.9%

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

   8. add-sqr-sqrt 17.8%

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

   9. sqrt-unprod 11.0%

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

   10. sqr-neg 11.0%

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

   11. sqrt-unprod 1.2%

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

   12. add-sqr-sqrt 2.1%

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

9. Applied egg-rr 2.1%

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

10. Taylor expanded in y around 0 2.3%

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

11. Final simplification 2.3%

    \[\leadsto t \]

Developer target: 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 2023308 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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