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

Percentage Accurate: 84.7% → 99.5%

Time: 10.7s

Alternatives: 8

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: 84.7% 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: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, \left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot y\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (fma (log y) x (* (* z (fma -0.5 y -1.0)) y)) t))

C

double code(double x, double y, double z, double t) {
	return fma(log(y), x, ((z * fma(-0.5, y, -1.0)) * y)) - t;
}

Julia

function code(x, y, z, t)
	return Float64(fma(log(y), x, Float64(Float64(z * fma(-0.5, y, -1.0)) * y)) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[(z * N[(-0.5 * y + -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. log-rec N/A

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

   6. lower-fma.f64 N/A

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

   7. log-rec N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. mul-1-neg N/A

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

   10. remove-double-neg N/A

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

   11. lower-log.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. associate-*r* N/A

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

   16. distribute-rgt-out N/A

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

   17. lower-*.f64 N/A

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

   18. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{-105} \lor \neg \left(t \leq 9.5 \cdot 10^{-62}\right):\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (or (<= t -8.8e-105) (not (<= t 9.5e-62)))
     (- t_1 t)
     (fma (- z) y t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((t <= -8.8e-105) || !(t <= 9.5e-62)) {
		tmp = t_1 - t;
	} else {
		tmp = fma(-z, y, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if ((t <= -8.8e-105) || !(t <= 9.5e-62))
		tmp = Float64(t_1 - t);
	else
		tmp = fma(Float64(-z), y, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[Or[LessEqual[t, -8.8e-105], N[Not[LessEqual[t, 9.5e-62]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[((-z) * y + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.80000000000000016e-105 or 9.49999999999999951e-62 < t

   1. Initial program 89.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. log-rec N/A

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

      6. lower-fma.f64 N/A

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

      7. log-rec N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. lower-log.f64 89.4

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

   8. Applied rewrites 89.4%

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

   if -8.80000000000000016e-105 < t < 9.49999999999999951e-62

   1. Initial program 74.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 95.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-105} \lor \neg \left(t \leq 9.5 \cdot 10^{-62}\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, \log y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+186}:\\ \;\;\;\;-\mathsf{fma}\left(z, y, t\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+175}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8e+186)
   (- (fma z y t))
   (if (<= z 5.8e+175)
     (- (* (log y) x) t)
     (- (* (* z (fma -0.5 y -1.0)) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8e+186) {
		tmp = -fma(z, y, t);
	} else if (z <= 5.8e+175) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = ((z * fma(-0.5, y, -1.0)) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8e+186)
		tmp = Float64(-fma(z, y, t));
	elseif (z <= 5.8e+175)
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(Float64(Float64(z * fma(-0.5, y, -1.0)) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8e+186], (-N[(z * y + t), $MachinePrecision]), If[LessEqual[z, 5.8e+175], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(z * N[(-0.5 * y + -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.99999999999999984e186

   1. Initial program 36.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 87.5%

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

   if -7.99999999999999984e186 < z < 5.8e175

   1. Initial program 92.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. log-rec N/A

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

      6. lower-fma.f64 N/A

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

      7. log-rec N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. lower-log.f64 92.4

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

   8. Applied rewrites 92.4%

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

   if 5.8e175 < z

   1. Initial program 53.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.9%

      \[\leadsto \left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot \color{blue}{y} - t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub N/A

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

   10. distribute-lft-neg-out N/A

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

   11. *-commutative N/A

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

   12. associate--l- N/A

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

   13. lower--.f64 N/A

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

5. Applied rewrites 99.8%

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

Alternative 5: 48.7% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-105} \lor \neg \left(t \leq 1.55 \cdot 10^{-76}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -8.8e-105) (not (<= t 1.55e-76))) (- t) (- (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -8.8e-105) || !(t <= 1.55e-76)) {
		tmp = -t;
	} else {
		tmp = -(z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-8.8d-105)) .or. (.not. (t <= 1.55d-76))) then
        tmp = -t
    else
        tmp = -(z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -8.8e-105) || !(t <= 1.55e-76)) {
		tmp = -t;
	} else {
		tmp = -(z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -8.8e-105) or not (t <= 1.55e-76):
		tmp = -t
	else:
		tmp = -(z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -8.8e-105) || !(t <= 1.55e-76))
		tmp = Float64(-t);
	else
		tmp = Float64(-Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -8.8e-105) || ~((t <= 1.55e-76)))
		tmp = -t;
	else
		tmp = -(z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -8.8e-105], N[Not[LessEqual[t, 1.55e-76]], $MachinePrecision]], (-t), (-N[(z * y), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if t < -8.80000000000000016e-105 or 1.54999999999999985e-76 < t

   1. Initial program 89.3%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

      2. lower-neg.f64 57.2

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

   5. Applied rewrites 57.2%

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

   if -8.80000000000000016e-105 < t < 1.54999999999999985e-76

   1. Initial program 74.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.4%

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

   9. Taylor expanded in y around inf

      \[\leadsto -y \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 28.3%

       \[\leadsto -z \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-105} \lor \neg \left(t \leq 1.55 \cdot 10^{-76}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot y - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* (* z (fma -0.5 y -1.0)) y) t))

C

double code(double x, double y, double z, double t) {
	return ((z * fma(-0.5, y, -1.0)) * y) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(z * fma(-0.5, y, -1.0)) * y) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(z * N[(-0.5 * y + -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 38.9

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

5. Applied rewrites 38.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 54.5%

   \[\leadsto \left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot \color{blue}{y} - t \]
9. Add Preprocessing

Alternative 7: 57.4% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub N/A

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

   10. distribute-lft-neg-out N/A

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

   11. *-commutative N/A

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

   12. associate--l- N/A

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

   13. lower--.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 54.5%

   \[\leadsto -\mathsf{fma}\left(z, y, t\right) \]
9. Add Preprocessing

Alternative 8: 42.4% accurate, 73.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. lower-neg.f64 38.7

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

5. Applied rewrites 38.7%

   \[\leadsto \color{blue}{-t} \]
6. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.3× 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 2024320 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

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