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

Percentage Accurate: 85.1% → 99.8%

Time: 12.8s

Alternatives: 10

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: 85.1% 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

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

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 81.8%

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

   1. +-commutative 81.8%

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

   2. fma-def 81.8%

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

   3. sub-neg 81.8%

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

   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 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in y around 0 99.5%

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

   1. fma-def 99.5%

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

   2. unpow2 99.5%

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

   3. associate-*r* 99.5%

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

   4. mul-1-neg 99.5%

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

4. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in y around 0 99.5%

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

   1. fma-def 99.5%

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

   2. unpow2 99.5%

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

   3. associate-*r* 99.5%

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

   4. mul-1-neg 99.5%

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

4. Simplified 99.5%

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

5. Taylor expanded in y around 0 98.8%

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

   1. remove-double-neg 98.8%

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

   2. log-rec 98.8%

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

   3. distribute-lft-neg-in 98.8%

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

   4. unsub-neg 98.8%

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

   5. mul-1-neg 98.8%

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

   6. distribute-rgt-neg-in 98.8%

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

   7. mul-1-neg 98.8%

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

   8. fma-neg 98.8%

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

   9. mul-1-neg 98.8%

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

   10. distribute-lft-neg-in 98.8%

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

   11. log-rec 98.8%

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

   12. remove-double-neg 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 90.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-84} \lor \neg \left(x \leq 9 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.5e-84) (not (<= x 9e-20)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.5e-84) || !(x <= 9e-20)) {
		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 <= -8.5e-84) || !(x <= 9e-20)) {
		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 <= -8.5e-84) or not (x <= 9e-20):
		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 <= -8.5e-84) || !(x <= 9e-20))
		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, -8.5e-84], N[Not[LessEqual[x, 9e-20]], $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 < -8.4999999999999994e-84 or 9.0000000000000003e-20 < x

   1. Initial program 91.1%

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

      1. +-commutative 91.1%

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

      2. associate--l+ 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-+l- 91.1%

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

      5. fma-neg 91.0%

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

      6. sub0-neg 91.0%

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

      7. associate-+l- 91.0%

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

      8. neg-sub0 91.0%

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

      9. +-commutative 91.0%

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

      10. fma-def 91.0%

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

      11. sub-neg 91.0%

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

      12. log1p-def 99.7%

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

   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 90.7%

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

   if -8.4999999999999994e-84 < x < 9.0000000000000003e-20

   1. Initial program 72.0%

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

   2. Taylor expanded in x around 0 61.5%

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

      1. sub-neg 61.5%

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

      2. mul-1-neg 61.5%

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

      3. log1p-def 89.3%

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

      4. mul-1-neg 89.3%

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

   4. Simplified 89.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-84} \lor \neg \left(x \leq 9 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 5: 90.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-84} \lor \neg \left(x \leq 3.7 \cdot 10^{-24}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(z \cdot -0.5\right) - z \cdot y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.15e-84) (not (<= x 3.7e-24)))
   (- (* x (log y)) t)
   (- (- (* (* y y) (* z -0.5)) (* z y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.15e-84) || !(x <= 3.7e-24)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (((y * y) * (z * -0.5)) - (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 <= (-1.15d-84)) .or. (.not. (x <= 3.7d-24))) then
        tmp = (x * log(y)) - t
    else
        tmp = (((y * y) * (z * (-0.5d0))) - (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 <= -1.15e-84) || !(x <= 3.7e-24)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (((y * y) * (z * -0.5)) - (z * y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.15e-84) or not (x <= 3.7e-24):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (((y * y) * (z * -0.5)) - (z * y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.15e-84) || !(x <= 3.7e-24))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * Float64(z * -0.5)) - Float64(z * y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.15e-84) || ~((x <= 3.7e-24)))
		tmp = (x * log(y)) - t;
	else
		tmp = (((y * y) * (z * -0.5)) - (z * y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.15e-84], N[Not[LessEqual[x, 3.7e-24]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999e-84 or 3.69999999999999981e-24 < x

   1. Initial program 91.1%

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

      1. +-commutative 91.1%

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

      2. associate--l+ 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-+l- 91.1%

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

      5. fma-neg 91.0%

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

      6. sub0-neg 91.0%

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

      7. associate-+l- 91.0%

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

      8. neg-sub0 91.0%

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

      9. +-commutative 91.0%

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

      10. fma-def 91.0%

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

      11. sub-neg 91.0%

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

      12. log1p-def 99.7%

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

   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 90.7%

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

   if -1.1499999999999999e-84 < x < 3.69999999999999981e-24

   1. Initial program 72.0%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. fma-def 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*r* 99.2%

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

      4. mul-1-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around 0 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*l* 88.6%

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

      3. fma-def 88.6%

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

      4. mul-1-neg 88.6%

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

      5. fma-neg 88.6%

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

      6. unpow2 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-84} \lor \neg \left(x \leq 3.7 \cdot 10^{-24}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(z \cdot -0.5\right) - z \cdot y\right) - t\\ \end{array} \]

Alternative 6: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in y around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. fma-def 98.8%

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

   3. mul-1-neg 98.8%

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

   4. fma-neg 98.8%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 7: 56.9% accurate, 16.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((y * y) * (z * -0.5)) - (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 = (((y * y) * (z * (-0.5d0))) - (z * y)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in y around 0 99.5%

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

   1. fma-def 99.5%

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

   2. unpow2 99.5%

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

   3. associate-*r* 99.5%

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

   4. mul-1-neg 99.5%

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

4. Simplified 99.5%

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

5. Taylor expanded in x around 0 63.2%

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

   1. *-commutative 63.2%

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

   2. associate-*l* 63.2%

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

   3. fma-def 63.2%

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

   4. mul-1-neg 63.2%

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

   5. fma-neg 63.2%

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

   6. unpow2 63.2%

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

7. Simplified 63.2%

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

8. Final simplification 63.2%

   \[\leadsto \left(\left(y \cdot y\right) \cdot \left(z \cdot -0.5\right) - z \cdot y\right) - t \]

Alternative 8: 56.9% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in y around 0 99.5%

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

   1. fma-def 99.5%

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

   2. unpow2 99.5%

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

   3. associate-*r* 99.5%

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

   4. mul-1-neg 99.5%

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

4. Simplified 99.5%

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

5. Taylor expanded in x around 0 63.2%

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

   1. associate-*r* 63.2%

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

   2. associate-*r* 63.2%

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

   3. distribute-rgt-in 63.2%

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

   4. mul-1-neg 63.2%

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

   5. unsub-neg 63.2%

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

   6. *-commutative 63.2%

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

   7. unpow2 63.2%

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

   8. associate-*l* 63.2%

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

7. Simplified 63.2%

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

8. Final simplification 63.2%

   \[\leadsto z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t \]

Alternative 9: 56.7% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. +-commutative 81.8%

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

   2. associate--l+ 81.8%

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

   3. +-commutative 81.8%

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

   4. associate-+l- 81.8%

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

   5. fma-neg 81.8%

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

   6. sub0-neg 81.8%

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

   7. associate-+l- 81.8%

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

   8. neg-sub0 81.8%

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

   9. +-commutative 81.8%

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

   10. fma-def 81.8%

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

   11. sub-neg 81.8%

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

   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 98.8%

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

   1. mul-1-neg 98.8%

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

   2. +-commutative 98.8%

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

   3. unsub-neg 98.8%

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

   4. mul-1-neg 98.8%

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

   5. distribute-rgt-neg-in 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in x around 0 62.6%

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

   1. associate-*r* 62.6%

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

   2. mul-1-neg 62.6%

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

9. Simplified 62.6%

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

10. Final simplification 62.6%

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

Alternative 10: 41.9% accurate, 105.5× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 81.8%

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

   1. +-commutative 81.8%

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

   2. associate--l+ 81.8%

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

   3. +-commutative 81.8%

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

   4. associate-+l- 81.8%

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

   5. fma-neg 81.8%

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

   6. sub0-neg 81.8%

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

   7. associate-+l- 81.8%

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

   8. neg-sub0 81.8%

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

   9. +-commutative 81.8%

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

   10. fma-def 81.8%

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

   11. sub-neg 81.8%

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

   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 t around inf 44.5%

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

   1. mul-1-neg 44.5%

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

6. Simplified 44.5%

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

7. Final simplification 44.5%

   \[\leadsto -t \]

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 2023229 
(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))