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

Percentage Accurate: 85.2% → 99.8%

Time: 14.7s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 90.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - t\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{-145}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 t)))
   (if (<= t -1.12e-145) t_2 (if (<= t 3.9e-75) (fma (- y) z t_1) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - t;
	double tmp;
	if (t <= -1.12e-145) {
		tmp = t_2;
	} else if (t <= 3.9e-75) {
		tmp = fma(-y, z, t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - t)
	tmp = 0.0
	if (t <= -1.12e-145)
		tmp = t_2;
	elseif (t <= 3.9e-75)
		tmp = fma(Float64(-y), z, t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - t), $MachinePrecision]}, If[LessEqual[t, -1.12e-145], t$95$2, If[LessEqual[t, 3.9e-75], N[((-y) * z + t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.12000000000000001e-145 or 3.9000000000000001e-75 < t

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if -1.12000000000000001e-145 < t < 3.9000000000000001e-75

   1. Initial program 67.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 93.0

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

   10. Applied rewrites 93.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-log.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 90.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-59}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -1.02e-141)
     t_1
     (if (<= x 2.5e-59) (- (* z (log1p (- y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -1.02e-141) {
		tmp = t_1;
	} else if (x <= 2.5e-59) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -1.02e-141) {
		tmp = t_1;
	} else if (x <= 2.5e-59) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -1.02e-141:
		tmp = t_1
	elif x <= 2.5e-59:
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -1.02e-141)
		tmp = t_1;
	elseif (x <= 2.5e-59)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.02e-141], t$95$1, If[LessEqual[x, 2.5e-59], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02e-141 or 2.5000000000000001e-59 < x

   1. Initial program 92.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -1.02e-141 < x < 2.5000000000000001e-59

   1. Initial program 74.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
   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. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-59}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -1.02e-141)
     t_1
     (if (<= x 2.5e-59) (- (* (* (fma -0.5 y -1.0) z) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -1.02e-141) {
		tmp = t_1;
	} else if (x <= 2.5e-59) {
		tmp = ((fma(-0.5, y, -1.0) * z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -1.02e-141)
		tmp = t_1;
	elseif (x <= 2.5e-59)
		tmp = Float64(Float64(Float64(fma(-0.5, y, -1.0) * z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.02e-141], t$95$1, If[LessEqual[x, 2.5e-59], N[(N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02e-141 or 2.5000000000000001e-59 < x

   1. Initial program 92.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -1.02e-141 < x < 2.5000000000000001e-59

   1. Initial program 74.5%

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 91.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+97}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -2.05e+110)
     t_1
     (if (<= x 5.4e+97) (- (* (* (fma -0.5 y -1.0) z) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -2.05e+110) {
		tmp = t_1;
	} else if (x <= 5.4e+97) {
		tmp = ((fma(-0.5, y, -1.0) * z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -2.05e+110)
		tmp = t_1;
	elseif (x <= 5.4e+97)
		tmp = Float64(Float64(Float64(fma(-0.5, y, -1.0) * z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.05e+110], t$95$1, If[LessEqual[x, 5.4e+97], N[(N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0499999999999999e110 or 5.39999999999999987e97 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -2.0499999999999999e110 < x < 5.39999999999999987e97

   1. Initial program 79.7%

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.8%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+97}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 71.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \left(-x\right) \cdot \log y\right)}{{\left(\mathsf{log1p}\left(-y\right) \cdot z\right)}^{2} - {\left(\log y \cdot x\right)}^{2}}}} - t \]

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. sub-neg N/A

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

   9. lower-log1p.f64 N/A

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

   10. lower-neg.f64 88.1

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

7. Applied rewrites 88.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 87.7%

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

11. Taylor expanded in y around 0

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

    1. associate--r+ N/A

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

    2. sub-neg N/A

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

    3. sub-neg N/A

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

    4. mul-1-neg N/A

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

    5. *-commutative N/A

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

    6. distribute-lft-neg-in N/A

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

    7. log-rec N/A

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

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

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

    9. log-rec N/A

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

    10. remove-double-neg N/A

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

    11. *-commutative N/A

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

    12. associate-+r+ N/A

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

    13. +-commutative N/A

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

    14. associate-+r+ N/A

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

    15. +-commutative N/A

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

    16. associate-+l+ N/A

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

    17. *-commutative N/A

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

    18. sub-neg N/A

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

    19. lower-fma.f64 N/A

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

13. Applied rewrites 99.5%

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

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate--l- N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 9: 48.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-145}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-75}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.12e-145) (- t) (if (<= t 3.9e-75) (* (- z) y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.12e-145) {
		tmp = -t;
	} else if (t <= 3.9e-75) {
		tmp = -z * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.12d-145)) then
        tmp = -t
    else if (t <= 3.9d-75) then
        tmp = -z * y
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.12e-145) {
		tmp = -t;
	} else if (t <= 3.9e-75) {
		tmp = -z * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.12e-145:
		tmp = -t
	elif t <= 3.9e-75:
		tmp = -z * y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.12e-145)
		tmp = Float64(-t);
	elseif (t <= 3.9e-75)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.12e-145)
		tmp = -t;
	elseif (t <= 3.9e-75)
		tmp = -z * y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.12e-145], (-t), If[LessEqual[t, 3.9e-75], N[((-z) * y), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.12000000000000001e-145 or 3.9000000000000001e-75 < t

   1. Initial program 94.6%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

   if -1.12000000000000001e-145 < t < 3.9000000000000001e-75

   1. Initial program 67.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--l- N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 34.6%

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

Alternative 10: 57.5% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-log.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 59.0%

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

Alternative 11: 57.2% 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 85.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate--l- N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

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 58.8%

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

Alternative 12: 42.7% 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 85.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 45.2

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

5. Applied rewrites 45.2%

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

Alternative 13: 2.3% accurate, 220.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 85.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 45.2

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

5. Applied rewrites 45.2%

   \[\leadsto \color{blue}{-t} \]
6. Step-by-step derivation

7. Applied rewrites 11.3%

   \[\leadsto \frac{0 - \left(t \cdot t\right) \cdot t}{\color{blue}{0 + \mathsf{fma}\left(t, t, 0 \cdot t\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 2.2%

   \[\leadsto t \]
10. Add Preprocessing

Developer Target 1: 99.5% 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 2024235 
(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))