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

Percentage Accurate: 85.2% → 99.3%

Time: 16.3s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(-y\right)\\ t_2 := \mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(x, \log y, t\right)\right)\\ t\_2 \cdot \left(\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, t\_1, -t\right)\right) \cdot \frac{1}{t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (log1p (- y))) (t_2 (fma z t_1 (fma x (log y) t))))
   (* t_2 (* (fma x (log y) (fma z t_1 (- t))) (/ 1.0 t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log1p(-y);
	double t_2 = fma(z, t_1, fma(x, log(y), t));
	return t_2 * (fma(x, log(y), fma(z, t_1, -t)) * (1.0 / t_2));
}

Julia

function code(x, y, z, t)
	t_1 = log1p(Float64(-y))
	t_2 = fma(z, t_1, fma(x, log(y), t))
	return Float64(t_2 * Float64(fma(x, log(y), fma(z, t_1, Float64(-t))) * Float64(1.0 / t_2)))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Log[1 + (-y)], $MachinePrecision]}, Block[{t$95$2 = N[(z * t$95$1 + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, N[(t$95$2 * N[(N[(x * N[Log[y], $MachinePrecision] + N[(z * t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 87.4%

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

   1. flip-- N/A

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

   2. div-inv N/A

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

   3. difference-of-squares N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)\right)}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (pow (/ 1.0 (fma x (log y) (fma z (log1p (- y)) (- t)))) -1.0))

C

double code(double x, double y, double z, double t) {
	return pow((1.0 / fma(x, log(y), fma(z, log1p(-y), -t))), -1.0);
}

Julia

function code(x, y, z, t)
	return Float64(1.0 / fma(x, log(y), fma(z, log1p(Float64(-y)), Float64(-t)))) ^ -1.0
end

Wolfram

code[x_, y_, z_, t_] := N[Power[N[(1.0 / N[(x * N[Log[y], $MachinePrecision] + N[(z * N[Log[1 + (-y)], $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-lowering-pow.f64 N/A

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

4. Applied egg-rr 99.6%

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

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)\right)}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ 1.0 (/ 1.0 (fma x (log y) (fma z (log1p (- y)) (- t))))))

C

double code(double x, double y, double z, double t) {
	return 1.0 / (1.0 / fma(x, log(y), fma(z, log1p(-y), -t)));
}

Julia

function code(x, y, z, t)
	return Float64(1.0 / Float64(1.0 / fma(x, log(y), fma(z, log1p(Float64(-y)), Float64(-t)))))
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 / N[(1.0 / N[(x * N[Log[y], $MachinePrecision] + N[(z * N[Log[1 + (-y)], $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-- N/A

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

   6. /-lowering-/.f64 N/A

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

   7. associate--l+ N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)\right)}}} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*r* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. sub-neg N/A

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

   17. remove-double-neg N/A

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

   18. mul-1-neg N/A

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

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

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

5. Simplified 99.4%

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

Alternative 5: 90.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) (- t))))
   (if (<= x -2.4e-83)
     t_1
     (if (<= x 1.22e-48) (fma z (log1p (- y)) (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, log(y), -t);
	double tmp;
	if (x <= -2.4e-83) {
		tmp = t_1;
	} else if (x <= 1.22e-48) {
		tmp = fma(z, log1p(-y), -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, log(y), Float64(-t))
	tmp = 0.0
	if (x <= -2.4e-83)
		tmp = t_1;
	elseif (x <= 1.22e-48)
		tmp = fma(z, log1p(Float64(-y)), Float64(-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] + (-t)), $MachinePrecision]}, If[LessEqual[x, -2.4e-83], t$95$1, If[LessEqual[x, 1.22e-48], N[(z * N[Log[1 + (-y)], $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e-83 or 1.21999999999999993e-48 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-rec N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. log-lowering-log.f64 N/A

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

      16. neg-lowering-neg.f64 93.1

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

   5. Simplified 93.1%

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

   if -2.4000000000000001e-83 < x < 1.21999999999999993e-48

   1. Initial program 77.7%

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

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. neg-lowering-neg.f64 91.9

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

   5. Simplified 91.9%

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

Alternative 6: 90.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, -t\right)\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \mathsf{fma}\left(y, -0.5, -1\right), -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) (- t))))
   (if (<= x -8.2e-84)
     t_1
     (if (<= x 1.95e-48) (fma (* z y) (fma y -0.5 -1.0) (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, log(y), -t);
	double tmp;
	if (x <= -8.2e-84) {
		tmp = t_1;
	} else if (x <= 1.95e-48) {
		tmp = fma((z * y), fma(y, -0.5, -1.0), -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, log(y), Float64(-t))
	tmp = 0.0
	if (x <= -8.2e-84)
		tmp = t_1;
	elseif (x <= 1.95e-48)
		tmp = fma(Float64(z * y), fma(y, -0.5, -1.0), Float64(-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] + (-t)), $MachinePrecision]}, If[LessEqual[x, -8.2e-84], t$95$1, If[LessEqual[x, 1.95e-48], N[(N[(z * y), $MachinePrecision] * N[(y * -0.5 + -1.0), $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2000000000000001e-84 or 1.95e-48 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-rec N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. log-lowering-log.f64 N/A

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

      16. neg-lowering-neg.f64 93.1

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

   5. Simplified 93.1%

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

   if -8.2000000000000001e-84 < x < 1.95e-48

   1. Initial program 77.7%

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

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. sub-neg N/A

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

      17. remove-double-neg N/A

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

      18. mul-1-neg N/A

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

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

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 91.7

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

   8. Simplified 91.7%

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

Alternative 7: 76.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, \mathsf{fma}\left(y, -0.5, -1\right), -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -4.1e+166)
     t_1
     (if (<= x 3e-6) (fma (* z y) (fma y -0.5 -1.0) (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -4.1e+166) {
		tmp = t_1;
	} else if (x <= 3e-6) {
		tmp = fma((z * y), fma(y, -0.5, -1.0), -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 <= -4.1e+166)
		tmp = t_1;
	elseif (x <= 3e-6)
		tmp = fma(Float64(z * y), fma(y, -0.5, -1.0), Float64(-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, -4.1e+166], t$95$1, If[LessEqual[x, 3e-6], N[(N[(z * y), $MachinePrecision] * N[(y * -0.5 + -1.0), $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1000000000000003e166 or 3.0000000000000001e-6 < x

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. *-lowering-*.f64 N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. log-lowering-log.f64 76.5

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

   5. Simplified 76.5%

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

   if -4.1000000000000003e166 < x < 3.0000000000000001e-6

   1. Initial program 82.5%

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

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. sub-neg N/A

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

      17. remove-double-neg N/A

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

      18. mul-1-neg N/A

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

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

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 80.7

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

   8. Simplified 80.7%

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

Alternative 8: 99.3% 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 87.4%

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

3. Taylor expanded in y around 0

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

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

   5. mul-1-neg N/A

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

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

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

   11. --lowering--.f64 N/A

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

5. Simplified 99.2%

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

Alternative 9: 44.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -z \cdot y\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+248}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+114}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z y))))
   (if (<= z -1.95e+248) t_1 (if (<= z 1.8e+114) (- t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -(z * y);
	double tmp;
	if (z <= -1.95e+248) {
		tmp = t_1;
	} else if (z <= 1.8e+114) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(z * y)
    if (z <= (-1.95d+248)) then
        tmp = t_1
    else if (z <= 1.8d+114) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -(z * y);
	double tmp;
	if (z <= -1.95e+248) {
		tmp = t_1;
	} else if (z <= 1.8e+114) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -(z * y)
	tmp = 0
	if z <= -1.95e+248:
		tmp = t_1
	elif z <= 1.8e+114:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-Float64(z * y))
	tmp = 0.0
	if (z <= -1.95e+248)
		tmp = t_1;
	elseif (z <= 1.8e+114)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -(z * y);
	tmp = 0.0;
	if (z <= -1.95e+248)
		tmp = t_1;
	elseif (z <= 1.8e+114)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = (-N[(z * y), $MachinePrecision])}, If[LessEqual[z, -1.95e+248], t$95$1, If[LessEqual[z, 1.8e+114], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9499999999999999e248 or 1.8e114 < z

   1. Initial program 54.2%

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

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

      5. mul-1-neg N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. associate--l- N/A

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

      11. --lowering--.f64 N/A

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

   5. Simplified 98.0%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 46.6

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

   8. Simplified 46.6%

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

   if -1.9499999999999999e248 < z < 1.8e114

   1. Initial program 95.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. neg-lowering-neg.f64 51.9

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

   5. Simplified 51.9%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+248}:\\ \;\;\;\;-z \cdot y\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+114}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*r* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. sub-neg N/A

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

   17. remove-double-neg N/A

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

   18. mul-1-neg N/A

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

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

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 58.7

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

8. Simplified 58.7%

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

Alternative 11: 57.5% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in y around 0

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

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

   5. mul-1-neg N/A

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

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

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

   11. --lowering--.f64 N/A

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. +-commutative N/A

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

   4. accelerator-lowering-fma.f64 58.5

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

8. Simplified 58.5%

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

Alternative 12: 42.9% 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 87.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. neg-lowering-neg.f64 45.8

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

5. Simplified 45.8%

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

Developer Target 1: 99.7% 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 2024199 
(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))