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

Percentage Accurate: 85.4% → 99.8%

Time: 12.8s

Alternatives: 11

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: 85.4% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. +-commutative 84.8%

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

   2. associate--l+ 84.8%

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

   3. +-commutative 84.8%

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

   4. associate-+l- 84.8%

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

   5. fma-neg 84.8%

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

   6. sub0-neg 84.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

   7. associate-+l- 84.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

   8. neg-sub0 84.8%

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

   9. +-commutative 84.8%

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

   10. fma-def 84.8%

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

   11. sub-neg 84.8%

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

   12. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. +-commutative 84.8%

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

   2. fma-def 84.8%

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

   3. sub-neg 84.8%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -1.06 \cdot 10^{-28} \lor \neg \left(t \leq 1.7 \cdot 10^{-60}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -1.06e-28) (not (<= t 1.7e-60))) (- t_1 t) (- t_1 (* y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((t <= -1.06e-28) || !(t <= 1.7e-60)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if ((t <= (-1.06d-28)) .or. (.not. (t <= 1.7d-60))) then
        tmp = t_1 - t
    else
        tmp = t_1 - (y * z)
    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 <= -1.06e-28) || !(t <= 1.7e-60)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (t <= -1.06e-28) or not (t <= 1.7e-60):
		tmp = t_1 - t
	else:
		tmp = t_1 - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((t <= -1.06e-28) || !(t <= 1.7e-60))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((t <= -1.06e-28) || ~((t <= 1.7e-60)))
		tmp = t_1 - t;
	else
		tmp = t_1 - (y * z);
	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, -1.06e-28], N[Not[LessEqual[t, 1.7e-60]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.06e-28 or 1.70000000000000003e-60 < t

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. fma-def 95.1%

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

      3. sub-neg 95.1%

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

      4. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 95.1%

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

   if -1.06e-28 < t < 1.70000000000000003e-60

   1. Initial program 74.5%

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

      1. +-commutative 74.5%

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

      2. associate--l+ 74.5%

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

      3. +-commutative 74.5%

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

      4. associate-+l- 74.5%

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

      5. fma-neg 74.5%

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

      6. sub0-neg 74.5%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

      7. associate-+l- 74.5%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

      8. neg-sub0 74.5%

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

      9. +-commutative 74.5%

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

      10. fma-def 74.5%

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

      11. sub-neg 74.5%

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

      12. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. +-commutative 99.1%

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

      3. unsub-neg 99.1%

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

      4. mul-1-neg 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in t around 0 88.2%

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

      1. mul-1-neg 88.2%

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

      2. sub-neg 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-28} \lor \neg \left(t \leq 1.7 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \]

Alternative 4: 89.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-113} \lor \neg \left(x \leq 3 \cdot 10^{-155}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.5e-113) (not (<= x 3e-155)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.5e-113) || !(x <= 3e-155)) {
		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 <= -2.5e-113) || !(x <= 3e-155)) {
		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 <= -2.5e-113) or not (x <= 3e-155):
		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 <= -2.5e-113) || !(x <= 3e-155))
		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, -2.5e-113], N[Not[LessEqual[x, 3e-155]], $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 < -2.4999999999999999e-113 or 2.99999999999999984e-155 < x

   1. Initial program 90.7%

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

      1. +-commutative 90.7%

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

      2. fma-def 90.7%

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

      3. sub-neg 90.7%

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

      4. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 90.1%

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

   if -2.4999999999999999e-113 < x < 2.99999999999999984e-155

   1. Initial program 73.2%

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

   2. Taylor expanded in x around 0 65.3%

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

      1. sub-neg 65.3%

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

      2. mul-1-neg 65.3%

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

      3. log1p-def 89.8%

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

      4. mul-1-neg 89.8%

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

   4. Simplified 89.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-113} \lor \neg \left(x \leq 3 \cdot 10^{-155}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 5: 89.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-113} \lor \neg \left(x \leq 1.1 \cdot 10^{-162}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.1e-113) (not (<= x 1.1e-162)))
   (- (* x (log y)) t)
   (- (- t) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.1e-113) || !(x <= 1.1e-162)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.1d-113)) .or. (.not. (x <= 1.1d-162))) then
        tmp = (x * log(y)) - t
    else
        tmp = -t - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.1e-113) || !(x <= 1.1e-162)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.1e-113) or not (x <= 1.1e-162):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -t - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.1e-113) || !(x <= 1.1e-162))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-t) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.1e-113) || ~((x <= 1.1e-162)))
		tmp = (x * log(y)) - t;
	else
		tmp = -t - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.1e-113], N[Not[LessEqual[x, 1.1e-162]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e-113 or 1.1e-162 < x

   1. Initial program 90.7%

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

      1. +-commutative 90.7%

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

      2. fma-def 90.7%

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

      3. sub-neg 90.7%

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

      4. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 90.1%

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

   if -2.1e-113 < x < 1.1e-162

   1. Initial program 73.2%

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

      1. +-commutative 73.2%

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

      2. associate--l+ 73.2%

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

      3. +-commutative 73.2%

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

      4. associate-+l- 73.2%

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

      5. fma-neg 73.2%

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

      6. sub0-neg 73.2%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

      7. associate-+l- 73.2%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

      8. neg-sub0 73.2%

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

      9. +-commutative 73.2%

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

      10. fma-def 73.2%

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

      11. sub-neg 73.2%

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

      12. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. unsub-neg 99.7%

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

      4. mul-1-neg 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 89.6%

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

      1. sub-neg 89.6%

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

      2. mul-1-neg 89.6%

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

      3. distribute-neg-out 89.6%

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

      4. add-sqr-sqrt 54.4%

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

      5. sqrt-unprod 72.7%

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

      6. sqr-neg 72.7%

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

      7. mul-1-neg 72.7%

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

      8. mul-1-neg 72.7%

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

      9. sqrt-unprod 43.7%

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

      10. add-sqr-sqrt 64.3%

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

      11. +-commutative 64.3%

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

      12. add-sqr-sqrt 43.7%

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

      13. sqrt-unprod 72.7%

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

      14. mul-1-neg 72.7%

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

      15. mul-1-neg 72.7%

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

      16. sqr-neg 72.7%

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

      17. sqrt-unprod 54.4%

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

      18. add-sqr-sqrt 89.6%

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

   9. Applied egg-rr 89.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-113} \lor \neg \left(x \leq 1.1 \cdot 10^{-162}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 6: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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)) - (t + (y * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. +-commutative 84.8%

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

   2. associate--l+ 84.8%

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

   3. +-commutative 84.8%

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

   4. associate-+l- 84.8%

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

   5. fma-neg 84.8%

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

   6. sub0-neg 84.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

   7. associate-+l- 84.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

   8. neg-sub0 84.8%

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

   9. +-commutative 84.8%

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

   10. fma-def 84.8%

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

   11. sub-neg 84.8%

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

   12. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 99.5%

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

   1. mul-1-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. unsub-neg 99.5%

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

   4. mul-1-neg 99.5%

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

   5. distribute-rgt-neg-in 99.5%

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

6. Simplified 99.5%

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

   1. fma-udef 99.5%

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

   2. *-commutative 99.5%

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

   3. sub-neg 99.5%

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

   4. associate-+r+ 99.5%

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

   5. distribute-rgt-neg-out 99.5%

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

   6. sub-neg 99.5%

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

   7. associate-+l- 99.5%

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

   8. *-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in y around 0 99.5%

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

10. Final simplification 99.5%

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

Alternative 7: 77.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+114} \lor \neg \left(x \leq 1.25 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.08e+114) (not (<= x 1.25e+113)))
   (* x (log y))
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.08e+114) || !(x <= 1.25e+113)) {
		tmp = x * log(y);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.08e+114) || !(x <= 1.25e+113))
		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, -1.08e+114], N[Not[LessEqual[x, 1.25e+113]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -1.08000000000000004e114 or 1.25e113 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l- 99.6%

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

      5. fma-neg 99.6%

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

      6. sub0-neg 99.6%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

      7. associate-+l- 99.6%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

      8. neg-sub0 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-def 99.6%

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

      11. sub-neg 99.6%

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

      12. log1p-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. unsub-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 90.4%

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

   if -1.08000000000000004e114 < x < 1.25e113

   1. Initial program 79.0%

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

      1. +-commutative 79.0%

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

      2. associate--l+ 79.0%

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

      3. +-commutative 79.0%

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

      4. associate-+l- 79.0%

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

      5. fma-neg 79.0%

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

      6. sub0-neg 79.0%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

      7. associate-+l- 79.0%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

      8. neg-sub0 79.0%

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

      9. +-commutative 79.0%

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

      10. fma-def 79.0%

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

      11. sub-neg 79.0%

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

      12. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. unsub-neg 99.5%

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

      4. mul-1-neg 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 73.9%

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

      1. sub-neg 73.9%

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

      2. mul-1-neg 73.9%

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

      3. distribute-neg-in 73.9%

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

      4. fma-def 73.9%

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

   9. Simplified 73.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+114} \lor \neg \left(x \leq 1.25 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 8: 77.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+118} \lor \neg \left(x \leq 1.3 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.05e+118) (not (<= x 1.3e+113)))
   (* x (log y))
   (- (- t) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.05e+118) || !(x <= 1.3e+113)) {
		tmp = x * log(y);
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.05d+118)) .or. (.not. (x <= 1.3d+113))) then
        tmp = x * log(y)
    else
        tmp = -t - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.05e+118) || !(x <= 1.3e+113)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.05e+118) or not (x <= 1.3e+113):
		tmp = x * math.log(y)
	else:
		tmp = -t - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.05e+118) || !(x <= 1.3e+113))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.05e+118) || ~((x <= 1.3e+113)))
		tmp = x * log(y);
	else
		tmp = -t - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.05e+118], N[Not[LessEqual[x, 1.3e+113]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0499999999999999e118 or 1.3e113 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l- 99.6%

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

      5. fma-neg 99.6%

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

      6. sub0-neg 99.6%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

      7. associate-+l- 99.6%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

      8. neg-sub0 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-def 99.6%

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

      11. sub-neg 99.6%

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

      12. log1p-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. unsub-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 90.4%

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

   if -2.0499999999999999e118 < x < 1.3e113

   1. Initial program 79.0%

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

      1. +-commutative 79.0%

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

      2. associate--l+ 79.0%

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

      3. +-commutative 79.0%

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

      4. associate-+l- 79.0%

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

      5. fma-neg 79.0%

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

      6. sub0-neg 79.0%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

      7. associate-+l- 79.0%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

      8. neg-sub0 79.0%

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

      9. +-commutative 79.0%

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

      10. fma-def 79.0%

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

      11. sub-neg 79.0%

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

      12. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. unsub-neg 99.5%

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

      4. mul-1-neg 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 73.9%

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

      1. sub-neg 73.9%

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

      2. mul-1-neg 73.9%

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

      3. distribute-neg-out 73.9%

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

      4. add-sqr-sqrt 43.0%

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

      5. sqrt-unprod 59.9%

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

      6. sqr-neg 59.9%

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

      7. mul-1-neg 59.9%

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

      8. mul-1-neg 59.9%

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

      9. sqrt-unprod 36.8%

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

      10. add-sqr-sqrt 54.0%

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

      11. +-commutative 54.0%

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

      12. add-sqr-sqrt 36.8%

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

      13. sqrt-unprod 59.9%

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

      14. mul-1-neg 59.9%

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

      15. mul-1-neg 59.9%

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

      16. sqr-neg 59.9%

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

      17. sqrt-unprod 43.0%

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

      18. add-sqr-sqrt 73.9%

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

   9. Applied egg-rr 73.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+118} \lor \neg \left(x \leq 1.3 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 9: 47.8% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-37}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5e-37) (- t) (if (<= t 5.2e-6) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5e-37) {
		tmp = -t;
	} else if (t <= 5.2e-6) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-37)) then
        tmp = -t
    else if (t <= 5.2d-6) then
        tmp = z * -y
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5e-37) {
		tmp = -t;
	} else if (t <= 5.2e-6) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5e-37:
		tmp = -t
	elif t <= 5.2e-6:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5e-37)
		tmp = Float64(-t);
	elseif (t <= 5.2e-6)
		tmp = Float64(z * Float64(-y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -5e-37)
		tmp = -t;
	elseif (t <= 5.2e-6)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5e-37], (-t), If[LessEqual[t, 5.2e-6], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -4.9999999999999997e-37 or 5.20000000000000019e-6 < t

   1. Initial program 96.5%

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

      1. +-commutative 96.5%

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

      2. fma-def 96.5%

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

      3. sub-neg 96.5%

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

      4. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 73.4%

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

      1. mul-1-neg 73.4%

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

   6. Simplified 73.4%

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

   if -4.9999999999999997e-37 < t < 5.20000000000000019e-6

   1. Initial program 73.8%

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

      1. +-commutative 73.8%

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

      2. associate--l+ 73.8%

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

      3. +-commutative 73.8%

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

      4. associate-+l- 73.8%

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

      5. fma-neg 73.8%

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

      6. sub0-neg 73.8%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

      7. associate-+l- 73.8%

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

      8. neg-sub0 73.8%

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

      9. +-commutative 73.8%

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

      10. fma-def 73.8%

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

      11. sub-neg 73.8%

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

      12. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. +-commutative 99.1%

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

      3. unsub-neg 99.1%

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

      4. mul-1-neg 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in y around inf 27.3%

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

      1. associate-*r* 27.3%

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

      2. neg-mul-1 27.3%

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

   9. Simplified 27.3%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-37}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 10: 56.7% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

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 - (y * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. +-commutative 84.8%

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

   2. associate--l+ 84.8%

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

   3. +-commutative 84.8%

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

   4. associate-+l- 84.8%

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

   5. fma-neg 84.8%

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

   6. sub0-neg 84.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

   7. associate-+l- 84.8%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

   8. neg-sub0 84.8%

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

   9. +-commutative 84.8%

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

   10. fma-def 84.8%

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

   11. sub-neg 84.8%

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

   12. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 99.5%

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

   1. mul-1-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. unsub-neg 99.5%

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

   4. mul-1-neg 99.5%

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

   5. distribute-rgt-neg-in 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 56.3%

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

   1. sub-neg 56.3%

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

   2. mul-1-neg 56.3%

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

   3. distribute-neg-out 56.3%

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

   4. add-sqr-sqrt 33.0%

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

   5. sqrt-unprod 46.2%

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

   6. sqr-neg 46.2%

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

   7. mul-1-neg 46.2%

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

   8. mul-1-neg 46.2%

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

   9. sqrt-unprod 27.7%

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

   10. add-sqr-sqrt 42.0%

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

   11. +-commutative 42.0%

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

   12. add-sqr-sqrt 27.7%

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

   13. sqrt-unprod 46.2%

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

   14. mul-1-neg 46.2%

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

   15. mul-1-neg 46.2%

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

   16. sqr-neg 46.2%

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

   17. sqrt-unprod 33.0%

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

   18. add-sqr-sqrt 56.3%

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

9. Applied egg-rr 56.3%

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

10. Final simplification 56.3%

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

Alternative 11: 42.3% 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 84.8%

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

   1. +-commutative 84.8%

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

   2. fma-def 84.8%

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

   3. sub-neg 84.8%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 42.2%

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

   1. mul-1-neg 42.2%

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

6. Simplified 42.2%

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

7. Final simplification 42.2%

   \[\leadsto -t \]

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 2023171 
(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))