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

Percentage Accurate: 84.9% → 99.8%

Time: 14.6s

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: 84.9% 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 86.6%

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

   1. +-commutative 86.6%

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

   2. associate--l+ 86.6%

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

   3. fma-def 86.6%

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

   4. sub-neg 86.6%

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

   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: 89.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-13} \lor \neg \left(t \leq 8 \cdot 10^{-62}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -2.1e-13) (not (<= t 8e-62))) (- 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 <= -2.1e-13) || !(t <= 8e-62)) {
		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 <= (-2.1d-13)) .or. (.not. (t <= 8d-62))) 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 <= -2.1e-13) || !(t <= 8e-62)) {
		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 <= -2.1e-13) or not (t <= 8e-62):
		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 <= -2.1e-13) || !(t <= 8e-62))
		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 <= -2.1e-13) || ~((t <= 8e-62)))
		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, -2.1e-13], N[Not[LessEqual[t, 8e-62]], $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 < -2.09999999999999989e-13 or 8.0000000000000003e-62 < t

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0 97.7%

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

   if -2.09999999999999989e-13 < t < 8.0000000000000003e-62

   1. Initial program 75.8%

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

   2. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. unsub-neg 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in t around 0 92.5%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-13} \lor \neg \left(t \leq 8 \cdot 10^{-62}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \end{array} \]

Alternative 3: 88.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-222} \lor \neg \left(x \leq 1.15 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7.5e-222) (not (<= x 1.15e-68)))
   (- (* x (log y)) t)
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7.5e-222) || !(x <= 1.15e-68)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7.5e-222) || !(x <= 1.15e-68))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7.5e-222], N[Not[LessEqual[x, 1.15e-68]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000004e-222 or 1.14999999999999998e-68 < x

   1. Initial program 93.7%

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

   2. Taylor expanded in y around 0 93.2%

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

   if -7.5000000000000004e-222 < x < 1.14999999999999998e-68

   1. Initial program 61.7%

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

   2. Taylor expanded in y around 0 98.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
   3. 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. mul-1-neg 98.8%

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

      3. unsub-neg 98.8%

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

   4. Simplified 98.8%

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

   5. Taylor expanded in x around 0 93.7%

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

      1. mul-1-neg 93.7%

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

      2. +-commutative 93.7%

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

      3. fma-udef 93.8%

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

   7. Simplified 93.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-222} \lor \neg \left(x \leq 1.15 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 4: 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 86.6%

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

2. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. mul-1-neg 99.3%

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

   3. unsub-neg 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 5: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+15} \lor \neg \left(x \leq 1.06 \cdot 10^{+65}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.72e+15) (not (<= x 1.06e+65)))
   (* x (log y))
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.72e+15) || !(x <= 1.06e+65)) {
		tmp = x * log(y);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.72e+15) || !(x <= 1.06e+65))
		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, -1.72e+15], N[Not[LessEqual[x, 1.06e+65]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -1.72e15 or 1.06e65 < x

   1. Initial program 97.1%

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

      1. flip3-- 97.1%

         \[\leadsto \left(x \cdot \log y + z \cdot \log \color{blue}{\left(\frac{{1}^{3} - {y}^{3}}{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}\right)}\right) - t \]

      2. log-div 97.1%

         \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log \left({1}^{3} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)}\right) - t \]

      3. metadata-eval 97.1%

         \[\leadsto \left(x \cdot \log y + z \cdot \left(\log \left(\color{blue}{1} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      4. pow3 97.1%

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

      5. sub-neg 97.1%

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

      6. distribute-rgt-neg-out 97.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 97.1%

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

      9. sqr-neg 97.1%

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

      10. sqrt-unprod 97.1%

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

      11. add-sqr-sqrt 97.1%

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

      12. log1p-udef 97.1%

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

      13. pow3 97.1%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      14. metadata-eval 97.1%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      15. log1p-udef 99.7%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

      16. *-un-lft-identity 99.7%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

      17. fma-def 99.7%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right)\right) - t \]

   3. Applied egg-rr 99.7%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   4. Taylor expanded in x around inf 78.7%

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

   if -1.72e15 < x < 1.06e65

   1. Initial program 79.2%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around 0 81.5%

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

      1. mul-1-neg 81.5%

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

      2. +-commutative 81.5%

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

      3. fma-udef 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+15} \lor \neg \left(x \leq 1.06 \cdot 10^{+65}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 6: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -42000000000000 \lor \neg \left(x \leq 7.6 \cdot 10^{+66}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -42000000000000.0) (not (<= x 7.6e+66)))
   (* x (log y))
   (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -42000000000000.0) || !(x <= 7.6e+66)) {
		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 <= (-42000000000000.0d0)) .or. (.not. (x <= 7.6d+66))) 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 <= -42000000000000.0) || !(x <= 7.6e+66)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -42000000000000.0) or not (x <= 7.6e+66):
		tmp = x * math.log(y)
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -42000000000000.0) || !(x <= 7.6e+66))
		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 <= -42000000000000.0) || ~((x <= 7.6e+66)))
		tmp = x * log(y);
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -42000000000000.0], N[Not[LessEqual[x, 7.6e+66]], $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 < -4.2e13 or 7.6000000000000004e66 < x

   1. Initial program 97.1%

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

      1. flip3-- 97.1%

         \[\leadsto \left(x \cdot \log y + z \cdot \log \color{blue}{\left(\frac{{1}^{3} - {y}^{3}}{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}\right)}\right) - t \]

      2. log-div 97.1%

         \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log \left({1}^{3} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)}\right) - t \]

      3. metadata-eval 97.1%

         \[\leadsto \left(x \cdot \log y + z \cdot \left(\log \left(\color{blue}{1} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      4. pow3 97.1%

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

      5. sub-neg 97.1%

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

      6. distribute-rgt-neg-out 97.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 97.1%

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

      9. sqr-neg 97.1%

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

      10. sqrt-unprod 97.1%

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

      11. add-sqr-sqrt 97.1%

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

      12. log1p-udef 97.1%

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

      13. pow3 97.1%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left(\color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      14. metadata-eval 97.1%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

      15. log1p-udef 99.7%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

      16. *-un-lft-identity 99.7%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(y \cdot y + \color{blue}{y}\right)\right)\right) - t \]

      17. fma-def 99.7%

          \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, y\right)}\right)\right)\right) - t \]

   3. Applied egg-rr 99.7%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   4. Taylor expanded in x around inf 78.7%

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

   if -4.2e13 < x < 7.6000000000000004e66

   1. Initial program 79.2%

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

   2. Taylor expanded in x around 0 61.3%

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

   3. Taylor expanded in y around 0 81.5%

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

      1. associate-*r* 81.5%

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

      2. mul-1-neg 81.5%

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

   5. Simplified 81.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -42000000000000 \lor \neg \left(x \leq 7.6 \cdot 10^{+66}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]

Alternative 7: 48.1% accurate, 25.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-59} \lor \neg \left(t \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.9e-59) (not (<= t 3e-60))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.9e-59) || !(t <= 3e-60)) {
		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 <= (-3.9d-59)) .or. (.not. (t <= 3d-60))) 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 <= -3.9e-59) || !(t <= 3e-60)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.9e-59) or not (t <= 3e-60):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.9e-59) || !(t <= 3e-60))
		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 <= -3.9e-59) || ~((t <= 3e-60)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.9e-59], N[Not[LessEqual[t, 3e-60]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.90000000000000019e-59 or 3.00000000000000019e-60 < t

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf 75.8%

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

      1. neg-mul-1 75.8%

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

   4. Simplified 75.8%

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

   if -3.90000000000000019e-59 < t < 3.00000000000000019e-60

   1. Initial program 74.4%

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

   2. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. unsub-neg 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in y around inf 28.7%

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

      1. mul-1-neg 28.7%

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

      2. *-commutative 28.7%

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

      3. distribute-rgt-neg-in 28.7%

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

   7. Simplified 28.7%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-59} \lor \neg \left(t \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 8: 48.0% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.45 \cdot 10^{-59}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.45e-59) (- (* z y) t) (if (<= t 2.15e-61) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.45e-59) {
		tmp = (z * y) - t;
	} else if (t <= 2.15e-61) {
		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 <= (-3.45d-59)) then
        tmp = (z * y) - t
    else if (t <= 2.15d-61) 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 <= -3.45e-59) {
		tmp = (z * y) - t;
	} else if (t <= 2.15e-61) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.45e-59:
		tmp = (z * y) - t
	elif t <= 2.15e-61:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.45e-59)
		tmp = Float64(Float64(z * y) - t);
	elseif (t <= 2.15e-61)
		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 <= -3.45e-59)
		tmp = (z * y) - t;
	elseif (t <= 2.15e-61)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.45e-59], N[(N[(z * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 2.15e-61], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < -3.44999999999999991e-59

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 74.5%

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

   3. Taylor expanded in y around 0 74.7%

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

      1. associate-*r* 74.7%

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

      2. mul-1-neg 74.7%

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

   5. Simplified 74.7%

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

      1. sub-neg 74.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 73.7%

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

      4. sqr-neg 73.7%

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

      5. sqrt-unprod 73.8%

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

      6. add-sqr-sqrt 73.8%

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

   7. Applied egg-rr 73.8%

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

      1. sub-neg 73.8%

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

   9. Simplified 73.8%

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

   if -3.44999999999999991e-59 < t < 2.1500000000000002e-61

   1. Initial program 74.4%

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

   2. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. unsub-neg 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in y around inf 28.7%

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

      1. mul-1-neg 28.7%

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

      2. *-commutative 28.7%

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

      3. distribute-rgt-neg-in 28.7%

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

   7. Simplified 28.7%

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

   if 2.1500000000000002e-61 < t

   1. Initial program 97.1%

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

   2. Taylor expanded in t around inf 78.2%

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

      1. neg-mul-1 78.2%

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

   4. Simplified 78.2%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.45 \cdot 10^{-59}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 9: 56.8% 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 86.6%

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

2. Taylor expanded in x around 0 44.4%

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

3. Taylor expanded in y around 0 57.1%

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

   1. associate-*r* 57.1%

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

   2. mul-1-neg 57.1%

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

5. Simplified 57.1%

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

6. Final simplification 57.1%

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

Alternative 10: 42.0% 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 86.6%

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

2. Taylor expanded in t around inf 43.6%

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

   1. neg-mul-1 43.6%

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

4. Simplified 43.6%

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

5. Final simplification 43.6%

   \[\leadsto -t \]

Alternative 11: 2.3% 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 86.6%

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

2. Taylor expanded in y around 0 85.8%

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

   1. fma-neg 85.8%

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

4. Simplified 85.8%

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

   1. add-sqr-sqrt 44.2%

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

   2. pow2 44.2%

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

   3. add-sqr-sqrt 35.8%

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

   4. sqrt-unprod 35.6%

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

   5. sqr-neg 35.6%

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

   6. sqrt-unprod 8.8%

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

   7. add-sqr-sqrt 23.5%

      \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x, \log y, \color{blue}{t}\right)}\right)}^{2} \]

6. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x, \log y, t\right)}\right)}^{2}} \]

7. Taylor expanded in x around 0 2.2%

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

8. Final simplification 2.2%

   \[\leadsto t \]

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 2023318 
(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))