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

Percentage Accurate: 84.9% → 99.8%

Time: 12.2s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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

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

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 86.1%

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

   1. +-commutative 86.1%

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

   2. fma-def 86.1%

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

   3. sub-neg 86.1%

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

   4. log1p-def 99.8%

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

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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

2. Taylor expanded in y around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. log-pow 42.6%

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

   4. mul-1-neg 42.6%

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

   5. unsub-neg 42.6%

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

   6. log-pow 99.1%

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

   7. *-commutative 99.1%

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

4. Simplified 99.1%

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

5. Taylor expanded in y around inf 99.1%

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

   1. distribute-lft-out 99.1%

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

   2. mul-1-neg 99.1%

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

   3. fma-def 99.1%

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

   4. log-rec 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+64} \lor \neg \left(x \leq 1.55 \cdot 10^{-54} \lor \neg \left(x \leq 2.35 \cdot 10^{-15}\right) \land x \leq 9.5 \cdot 10^{+100}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.4e+64)
         (not
          (or (<= x 1.55e-54) (and (not (<= x 2.35e-15)) (<= x 9.5e+100)))))
   (* x (log y))
   (- (* z (- (* y (* y -0.5)) y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.4e+64) || !((x <= 1.55e-54) || (!(x <= 2.35e-15) && (x <= 9.5e+100)))) {
		tmp = x * log(y);
	} else {
		tmp = (z * ((y * (y * -0.5)) - 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 <= (-1.4d+64)) .or. (.not. (x <= 1.55d-54) .or. (.not. (x <= 2.35d-15)) .and. (x <= 9.5d+100))) then
        tmp = x * log(y)
    else
        tmp = (z * ((y * (y * (-0.5d0))) - 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 <= -1.4e+64) || !((x <= 1.55e-54) || (!(x <= 2.35e-15) && (x <= 9.5e+100)))) {
		tmp = x * Math.log(y);
	} else {
		tmp = (z * ((y * (y * -0.5)) - y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.4e+64) or not ((x <= 1.55e-54) or (not (x <= 2.35e-15) and (x <= 9.5e+100))):
		tmp = x * math.log(y)
	else:
		tmp = (z * ((y * (y * -0.5)) - y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.4e+64) || !((x <= 1.55e-54) || (!(x <= 2.35e-15) && (x <= 9.5e+100))))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(z * Float64(Float64(y * Float64(y * -0.5)) - y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.4e+64) || ~(((x <= 1.55e-54) || (~((x <= 2.35e-15)) && (x <= 9.5e+100)))))
		tmp = x * log(y);
	else
		tmp = (z * ((y * (y * -0.5)) - y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.4e+64], N[Not[Or[LessEqual[x, 1.55e-54], And[N[Not[LessEqual[x, 2.35e-15]], $MachinePrecision], LessEqual[x, 9.5e+100]]]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000012e64 or 1.55000000000000002e-54 < x < 2.3499999999999999e-15 or 9.4999999999999995e100 < x

   1. Initial program 95.6%

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

   2. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. log-pow 8.3%

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

      4. mul-1-neg 8.3%

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

      5. unsub-neg 8.3%

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

      6. log-pow 99.6%

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

      7. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. mul-1-neg 99.6%

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

      3. fma-def 99.6%

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

      4. log-rec 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 84.8%

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

   if -1.40000000000000012e64 < x < 1.55000000000000002e-54 or 2.3499999999999999e-15 < x < 9.4999999999999995e100

   1. Initial program 80.1%

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

      1. +-commutative 80.1%

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

      2. fma-def 80.1%

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

      3. sub-neg 80.1%

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

      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 x around 0 62.5%

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

      1. fma-neg 62.5%

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

      2. sub-neg 62.5%

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

      3. mul-1-neg 62.5%

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

      4. log1p-def 81.5%

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

      5. mul-1-neg 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in y around 0 80.7%

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

      1. mul-1-neg 80.7%

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

      2. +-commutative 80.7%

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

      3. associate-+r+ 80.7%

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

      4. associate-*r* 80.7%

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

      5. associate-*r* 80.7%

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

      6. distribute-rgt-out 80.7%

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

      7. mul-1-neg 80.7%

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

      8. unsub-neg 80.7%

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

      9. *-commutative 80.7%

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

      10. unpow2 80.7%

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

      11. associate-*l* 80.7%

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

   9. Simplified 80.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+64} \lor \neg \left(x \leq 1.55 \cdot 10^{-54} \lor \neg \left(x \leq 2.35 \cdot 10^{-15}\right) \land x \leq 9.5 \cdot 10^{+100}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t\\ \end{array} \]

Alternative 4: 89.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-62} \lor \neg \left(t \leq 4.5 \cdot 10^{-89}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -1.4e-62) (not (<= t 4.5e-89))) (- 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 <= -1.4e-62) || !(t <= 4.5e-89)) {
		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 <= (-1.4d-62)) .or. (.not. (t <= 4.5d-89))) 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 <= -1.4e-62) || !(t <= 4.5e-89)) {
		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 <= -1.4e-62) or not (t <= 4.5e-89):
		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 <= -1.4e-62) || !(t <= 4.5e-89))
		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 <= -1.4e-62) || ~((t <= 4.5e-89)))
		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, -1.4e-62], N[Not[LessEqual[t, 4.5e-89]], $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 < -1.40000000000000001e-62 or 4.4999999999999999e-89 < t

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. fma-def 93.6%

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

      3. sub-neg 93.6%

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

      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 z around 0 92.8%

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

   if -1.40000000000000001e-62 < t < 4.4999999999999999e-89

   1. Initial program 71.9%

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

   2. Taylor expanded in y around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. log-pow 36.9%

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

      4. mul-1-neg 36.9%

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

      5. unsub-neg 36.9%

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

      6. log-pow 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in t around 0 89.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-62} \lor \neg \left(t \leq 4.5 \cdot 10^{-89}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \end{array} \]

Alternative 5: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-65} \lor \neg \left(x \leq 8.5 \cdot 10^{-216}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.2e-65) (not (<= x 8.5e-216)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.2e-65) || !(x <= 8.5e-216)) {
		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 <= -6.2e-65) || !(x <= 8.5e-216)) {
		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 <= -6.2e-65) or not (x <= 8.5e-216):
		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 <= -6.2e-65) || !(x <= 8.5e-216))
		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, -6.2e-65], N[Not[LessEqual[x, 8.5e-216]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000032e-65 or 8.50000000000000003e-216 < x

   1. Initial program 90.3%

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

      1. +-commutative 90.3%

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

      2. fma-def 90.3%

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

      3. sub-neg 90.3%

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

      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 89.9%

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

   if -6.20000000000000032e-65 < x < 8.50000000000000003e-216

   1. Initial program 72.8%

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

   2. Taylor expanded in x around 0 69.9%

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

      1. sub-neg 69.9%

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

      2. mul-1-neg 69.9%

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

      3. log1p-def 97.0%

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

      4. mul-1-neg 97.0%

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

   4. Simplified 97.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-65} \lor \neg \left(x \leq 8.5 \cdot 10^{-216}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 6: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-70} \lor \neg \left(x \leq 8 \cdot 10^{-216}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.8e-70) (not (<= x 8e-216)))
   (- (* x (log y)) t)
   (- (* z (- (* y (* y -0.5)) y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.8e-70) || !(x <= 8e-216)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * ((y * (y * -0.5)) - 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 <= (-6.8d-70)) .or. (.not. (x <= 8d-216))) then
        tmp = (x * log(y)) - t
    else
        tmp = (z * ((y * (y * (-0.5d0))) - 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 <= -6.8e-70) || !(x <= 8e-216)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * ((y * (y * -0.5)) - y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.8e-70) or not (x <= 8e-216):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * ((y * (y * -0.5)) - y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.8e-70) || !(x <= 8e-216))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(Float64(y * Float64(y * -0.5)) - y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -6.8e-70) || ~((x <= 8e-216)))
		tmp = (x * log(y)) - t;
	else
		tmp = (z * ((y * (y * -0.5)) - y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.8e-70], N[Not[LessEqual[x, 8e-216]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[(N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.79999999999999991e-70 or 8.0000000000000003e-216 < x

   1. Initial program 90.3%

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

      1. +-commutative 90.3%

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

      2. fma-def 90.3%

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

      3. sub-neg 90.3%

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

      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 89.9%

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

   if -6.79999999999999991e-70 < x < 8.0000000000000003e-216

   1. Initial program 72.8%

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

      1. +-commutative 72.8%

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

      2. fma-def 72.8%

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

      3. sub-neg 72.8%

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

      4. log1p-def 99.9%

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

   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 x around 0 69.9%

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

      1. fma-neg 69.9%

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

      2. sub-neg 69.9%

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

      3. mul-1-neg 69.9%

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

      4. log1p-def 97.1%

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

      5. mul-1-neg 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in y around 0 95.8%

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

      1. mul-1-neg 95.8%

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

      2. +-commutative 95.8%

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

      3. associate-+r+ 95.8%

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

      4. associate-*r* 95.8%

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

      5. associate-*r* 95.8%

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

      6. distribute-rgt-out 95.8%

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

      7. mul-1-neg 95.8%

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

      8. unsub-neg 95.8%

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

      9. *-commutative 95.8%

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

      10. unpow2 95.8%

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

      11. associate-*l* 95.8%

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

   9. Simplified 95.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-70} \lor \neg \left(x \leq 8 \cdot 10^{-216}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) - t\\ \end{array} \]

Alternative 7: 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 86.1%

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

2. Taylor expanded in y around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. log-pow 42.6%

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

   4. mul-1-neg 42.6%

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

   5. unsub-neg 42.6%

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

   6. log-pow 99.1%

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

   7. *-commutative 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 8: 57.7% 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 86.1%

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

   1. +-commutative 86.1%

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

   2. fma-def 86.1%

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

   3. sub-neg 86.1%

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

   4. log1p-def 99.8%

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

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 x around 0 43.1%

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

   1. fma-neg 43.1%

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

   2. sub-neg 43.1%

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

   3. mul-1-neg 43.1%

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

   4. log1p-def 56.0%

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

   5. mul-1-neg 56.0%

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

6. Simplified 56.0%

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

7. Taylor expanded in y around 0 55.6%

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

   1. mul-1-neg 55.6%

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

   2. +-commutative 55.6%

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

   3. associate-+r+ 55.6%

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

   4. associate-*r* 55.6%

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

   5. associate-*r* 55.6%

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

   6. distribute-rgt-out 55.6%

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

   7. mul-1-neg 55.6%

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

   8. unsub-neg 55.6%

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

   9. *-commutative 55.6%

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

   10. unpow2 55.6%

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

   11. associate-*l* 55.6%

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

9. Simplified 55.6%

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

10. Final simplification 55.6%

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

Alternative 9: 48.6% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-75}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-86}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.8e-75) (- t) (if (<= t 2.6e-86) (* y (- z)) (- (* z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.8e-75) {
		tmp = -t;
	} else if (t <= 2.6e-86) {
		tmp = y * -z;
	} 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 (t <= (-7.8d-75)) then
        tmp = -t
    else if (t <= 2.6d-86) then
        tmp = y * -z
    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 (t <= -7.8e-75) {
		tmp = -t;
	} else if (t <= 2.6e-86) {
		tmp = y * -z;
	} else {
		tmp = (z * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7.8e-75:
		tmp = -t
	elif t <= 2.6e-86:
		tmp = y * -z
	else:
		tmp = (z * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.8e-75)
		tmp = Float64(-t);
	elseif (t <= 2.6e-86)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(Float64(z * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7.8e-75)
		tmp = -t;
	elseif (t <= 2.6e-86)
		tmp = y * -z;
	else
		tmp = (z * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.8e-75], (-t), If[LessEqual[t, 2.6e-86], N[(y * (-z)), $MachinePrecision], N[(N[(z * y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.8000000000000003e-75

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. fma-def 91.0%

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

      3. sub-neg 91.0%

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

      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 55.1%

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

      1. mul-1-neg 55.1%

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

   6. Simplified 55.1%

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

   if -7.8000000000000003e-75 < t < 2.6000000000000001e-86

   1. Initial program 71.9%

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

   2. Taylor expanded in y around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. log-pow 36.4%

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

      4. mul-1-neg 36.4%

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

      5. unsub-neg 36.4%

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

      6. log-pow 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. mul-1-neg 98.9%

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

      3. fma-def 98.9%

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

      4. log-rec 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around inf 29.9%

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

      1. associate-*r* 29.9%

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

      2. neg-mul-1 29.9%

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

   10. Simplified 29.9%

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

   if 2.6000000000000001e-86 < t

   1. Initial program 96.1%

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

      1. *-commutative 96.1%

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

      2. add-cube-cbrt 95.7%

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

      3. associate-*l* 95.6%

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

      4. fma-def 95.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}, \sqrt[3]{\log y} \cdot x, z \cdot \log \left(1 - y\right)\right)} - t \]

      5. cbrt-unprod 95.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{\log y \cdot \log y}}, \sqrt[3]{\log y} \cdot x, z \cdot \log \left(1 - y\right)\right) - t \]

      6. pow2 95.8%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\color{blue}{{\log y}^{2}}}, \sqrt[3]{\log y} \cdot x, z \cdot \log \left(1 - y\right)\right) - t \]

      7. sub-neg 95.8%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, z \cdot \log \color{blue}{\left(1 + \left(-y\right)\right)}\right) - t \]

      8. log1p-udef 99.5%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)}\right) - t \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, z \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)\right) - t \]

      10. sqrt-unprod 95.8%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, z \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)\right) - t \]

      11. sqr-neg 95.8%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, z \cdot \mathsf{log1p}\left(\sqrt{\color{blue}{y \cdot y}}\right)\right) - t \]

      12. sqrt-unprod 95.7%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, z \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)\right) - t \]

      13. add-sqr-sqrt 95.7%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, z \cdot \mathsf{log1p}\left(\color{blue}{y}\right)\right) - t \]

   3. Applied egg-rr 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, z \cdot \mathsf{log1p}\left(y\right)\right)} - t \]

   4. Taylor expanded in y around 0 95.7%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log y}^{2}}, \sqrt[3]{\log y} \cdot x, \color{blue}{y \cdot z}\right) - t \]

   5. Taylor expanded in y around inf 61.4%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-75}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-86}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y - t\\ \end{array} \]

Alternative 10: 48.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-75}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-86}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6.2e-75) (- t) (if (<= t 3.3e-86) (* y (- z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.2e-75) {
		tmp = -t;
	} else if (t <= 3.3e-86) {
		tmp = y * -z;
	} 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 <= (-6.2d-75)) then
        tmp = -t
    else if (t <= 3.3d-86) then
        tmp = y * -z
    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 <= -6.2e-75) {
		tmp = -t;
	} else if (t <= 3.3e-86) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6.2e-75:
		tmp = -t
	elif t <= 3.3e-86:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6.2e-75)
		tmp = Float64(-t);
	elseif (t <= 3.3e-86)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -6.2e-75)
		tmp = -t;
	elseif (t <= 3.3e-86)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6.2e-75], (-t), If[LessEqual[t, 3.3e-86], N[(y * (-z)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -6.20000000000000013e-75 or 3.29999999999999987e-86 < t

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. fma-def 93.6%

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

      3. sub-neg 93.6%

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

      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 58.3%

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

      1. mul-1-neg 58.3%

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

   6. Simplified 58.3%

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

   if -6.20000000000000013e-75 < t < 3.29999999999999987e-86

   1. Initial program 71.9%

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

   2. Taylor expanded in y around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. log-pow 36.4%

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

      4. mul-1-neg 36.4%

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

      5. unsub-neg 36.4%

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

      6. log-pow 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. mul-1-neg 98.9%

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

      3. fma-def 98.9%

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

      4. log-rec 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around inf 29.9%

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

      1. associate-*r* 29.9%

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

      2. neg-mul-1 29.9%

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

   10. Simplified 29.9%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-75}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-86}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 57.4% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

2. Taylor expanded in y around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. log-pow 42.6%

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

   4. mul-1-neg 42.6%

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

   5. unsub-neg 42.6%

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

   6. log-pow 99.1%

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

   7. *-commutative 99.1%

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

4. Simplified 99.1%

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

5. Taylor expanded in y around inf 55.3%

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

   1. mul-1-neg 55.3%

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

   2. distribute-rgt-neg-out 55.3%

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

7. Simplified 55.3%

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

8. Final simplification 55.3%

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

Alternative 12: 42.5% 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 86.1%

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

   1. +-commutative 86.1%

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

   2. fma-def 86.1%

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

   3. sub-neg 86.1%

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

   4. log1p-def 99.8%

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

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: 99.5% 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 2023268 
(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))