Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 89.7% → 99.6%

Time: 15.8s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * 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 - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * 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 - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

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

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

   4. clear-num N/A

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

   5. flip-- N/A

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-1 + z\right) \cdot \log \left(1 - y\right) + \left(x + -1\right) \cdot \log y\\ \mathbf{if}\;t\_1 \leq -500000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t\_1 \leq 180:\\ \;\;\;\;\left(y - y \cdot z\right) - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (+ -1.0 z) (log (- 1.0 y))) (* (+ x -1.0) (log y)))))
   (if (<= t_1 -500000000.0)
     (- (* x (log y)) t)
     (if (<= t_1 180.0)
       (- (- y (* y z)) t)
       (if (<= t_1 1000.0) (- (- t) (log y)) (fma (log y) x (- t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((-1.0 + z) * log((1.0 - y))) + ((x + -1.0) * log(y));
	double tmp;
	if (t_1 <= -500000000.0) {
		tmp = (x * log(y)) - t;
	} else if (t_1 <= 180.0) {
		tmp = (y - (y * z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = -t - log(y);
	} else {
		tmp = fma(log(y), x, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-1.0 + z) * log(Float64(1.0 - y))) + Float64(Float64(x + -1.0) * log(y)))
	tmp = 0.0
	if (t_1 <= -500000000.0)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (t_1 <= 180.0)
		tmp = Float64(Float64(y - Float64(y * z)) - t);
	elseif (t_1 <= 1000.0)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = fma(log(y), x, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(-1.0 + z), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -500000000.0], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 180.0], N[(N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -5e8

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 93.9

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

   5. Simplified 93.9%

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

   if -5e8 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 180

   1. Initial program 60.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 75.9

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

   8. Simplified 75.9%

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

   if 180 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 1e3

   1. Initial program 92.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. neg-lowering-neg.f64 92.0

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

   7. Simplified 92.0%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

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

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

      7. log-lowering-log.f64 91.1

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

   10. Simplified 91.1%

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

   if 1e3 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 96.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. neg-lowering-neg.f64 95.6

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

   7. Simplified 95.6%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 93.7%

       \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{x}, -t\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 + z\right) \cdot \log \left(1 - y\right) + \left(x + -1\right) \cdot \log y \leq -500000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\left(-1 + z\right) \cdot \log \left(1 - y\right) + \left(x + -1\right) \cdot \log y \leq 180:\\ \;\;\;\;\left(y - y \cdot z\right) - t\\ \mathbf{elif}\;\left(-1 + z\right) \cdot \log \left(1 - y\right) + \left(x + -1\right) \cdot \log y \leq 1000:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ t_2 := \left(-1 + z\right) \cdot \log \left(1 - y\right) + \left(x + -1\right) \cdot \log y\\ \mathbf{if}\;t\_2 \leq -500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 180:\\ \;\;\;\;\left(y - y \cdot z\right) - t\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t))
        (t_2 (+ (* (+ -1.0 z) (log (- 1.0 y))) (* (+ x -1.0) (log y)))))
   (if (<= t_2 -500000000.0)
     t_1
     (if (<= t_2 180.0)
       (- (- y (* y z)) t)
       (if (<= t_2 1000.0) (- (- t) (log y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double t_2 = ((-1.0 + z) * log((1.0 - y))) + ((x + -1.0) * log(y));
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = t_1;
	} else if (t_2 <= 180.0) {
		tmp = (y - (y * z)) - t;
	} else if (t_2 <= 1000.0) {
		tmp = -t - log(y);
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (x * log(y)) - t
    t_2 = (((-1.0d0) + z) * log((1.0d0 - y))) + ((x + (-1.0d0)) * log(y))
    if (t_2 <= (-500000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 180.0d0) then
        tmp = (y - (y * z)) - t
    else if (t_2 <= 1000.0d0) then
        tmp = -t - log(y)
    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 = (x * Math.log(y)) - t;
	double t_2 = ((-1.0 + z) * Math.log((1.0 - y))) + ((x + -1.0) * Math.log(y));
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = t_1;
	} else if (t_2 <= 180.0) {
		tmp = (y - (y * z)) - t;
	} else if (t_2 <= 1000.0) {
		tmp = -t - Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	t_2 = ((-1.0 + z) * math.log((1.0 - y))) + ((x + -1.0) * math.log(y))
	tmp = 0
	if t_2 <= -500000000.0:
		tmp = t_1
	elif t_2 <= 180.0:
		tmp = (y - (y * z)) - t
	elif t_2 <= 1000.0:
		tmp = -t - math.log(y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	t_2 = Float64(Float64(Float64(-1.0 + z) * log(Float64(1.0 - y))) + Float64(Float64(x + -1.0) * log(y)))
	tmp = 0.0
	if (t_2 <= -500000000.0)
		tmp = t_1;
	elseif (t_2 <= 180.0)
		tmp = Float64(Float64(y - Float64(y * z)) - t);
	elseif (t_2 <= 1000.0)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - t;
	t_2 = ((-1.0 + z) * log((1.0 - y))) + ((x + -1.0) * log(y));
	tmp = 0.0;
	if (t_2 <= -500000000.0)
		tmp = t_1;
	elseif (t_2 <= 180.0)
		tmp = (y - (y * z)) - t;
	elseif (t_2 <= 1000.0)
		tmp = -t - log(y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-1.0 + z), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500000000.0], t$95$1, If[LessEqual[t$95$2, 180.0], N[(N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 1000.0], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -5e8 or 1e3 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 93.8

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

   5. Simplified 93.8%

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

   if -5e8 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 180

   1. Initial program 60.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 75.9

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

   8. Simplified 75.9%

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

   if 180 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 1e3

   1. Initial program 92.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. neg-lowering-neg.f64 92.0

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

   7. Simplified 92.0%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

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

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

      7. log-lowering-log.f64 91.1

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

   10. Simplified 91.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 + z\right) \cdot \log \left(1 - y\right) + \left(x + -1\right) \cdot \log y \leq -500000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\left(-1 + z\right) \cdot \log \left(1 - y\right) + \left(x + -1\right) \cdot \log y \leq 180:\\ \;\;\;\;\left(y - y \cdot z\right) - t\\ \mathbf{elif}\;\left(-1 + z\right) \cdot \log \left(1 - y\right) + \left(x + -1\right) \cdot \log y \leq 1000:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -1.0000001:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \mathbf{elif}\;x + -1 \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x -1.0) -1.0000001)
   (fma (log y) (+ x -1.0) (- t))
   (if (<= (+ x -1.0) 500000.0)
     (- (fma y (- 1.0 z) (- (log y))) t)
     (- (fma (log y) (+ x -1.0) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + -1.0) <= -1.0000001) {
		tmp = fma(log(y), (x + -1.0), -t);
	} else if ((x + -1.0) <= 500000.0) {
		tmp = fma(y, (1.0 - z), -log(y)) - t;
	} else {
		tmp = fma(log(y), (x + -1.0), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -1.0000001)
		tmp = fma(log(y), Float64(x + -1.0), Float64(-t));
	elseif (Float64(x + -1.0) <= 500000.0)
		tmp = Float64(fma(y, Float64(1.0 - z), Float64(-log(y))) - t);
	else
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + -1.0), $MachinePrecision], -1.0000001], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[N[(x + -1.0), $MachinePrecision], 500000.0], N[(N[(y * N[(1.0 - z), $MachinePrecision] + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.00000010000000006

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. neg-lowering-neg.f64 96.6

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

   5. Simplified 96.6%

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

   if -1.00000010000000006 < (-.f64 x #s(literal 1 binary64)) < 5e5

   1. Initial program 84.4%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. log-lowering-log.f64 99.3

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

   8. Simplified 99.3%

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

   if 5e5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 95.7

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

   8. Simplified 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x + -1, y\right)} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -1.0000001:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \mathbf{elif}\;x + -1 \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + z \leq -2 \cdot 10^{+241}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + z \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 z) -2e+241)
   (- (fma y z t))
   (if (<= (+ -1.0 z) 2e+224)
     (- (fma (log y) (+ x -1.0) y) t)
     (- (* z (* y (fma y -0.5 -1.0))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + z) <= -2e+241) {
		tmp = -fma(y, z, t);
	} else if ((-1.0 + z) <= 2e+224) {
		tmp = fma(log(y), (x + -1.0), y) - t;
	} else {
		tmp = (z * (y * fma(y, -0.5, -1.0))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + z) <= -2e+241)
		tmp = Float64(-fma(y, z, t));
	elseif (Float64(-1.0 + z) <= 2e+224)
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	else
		tmp = Float64(Float64(z * Float64(y * fma(y, -0.5, -1.0))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + z), $MachinePrecision], -2e+241], (-N[(y * z + t), $MachinePrecision]), If[LessEqual[N[(-1.0 + z), $MachinePrecision], 2e+224], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -2.0000000000000001e241

   1. Initial program 32.3%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 81.2

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

   5. Simplified 81.2%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. accelerator-lowering-fma.f64 81.2

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

   8. Simplified 81.2%

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

   if -2.0000000000000001e241 < (-.f64 z #s(literal 1 binary64)) < 1.99999999999999994e224

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 96.7

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

   8. Simplified 96.7%

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

   if 1.99999999999999994e224 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 52.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 76.7

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

   5. Simplified 76.7%

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

   6. Taylor expanded in y around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 76.7

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

   8. Simplified 76.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + z \leq -2 \cdot 10^{+241}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + z \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + z \leq -2 \cdot 10^{+241}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + z \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 z) -2e+241)
   (- (fma y z t))
   (if (<= (+ -1.0 z) 2e+224)
     (fma (log y) (+ x -1.0) (- t))
     (- (* z (* y (fma y -0.5 -1.0))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + z) <= -2e+241) {
		tmp = -fma(y, z, t);
	} else if ((-1.0 + z) <= 2e+224) {
		tmp = fma(log(y), (x + -1.0), -t);
	} else {
		tmp = (z * (y * fma(y, -0.5, -1.0))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + z) <= -2e+241)
		tmp = Float64(-fma(y, z, t));
	elseif (Float64(-1.0 + z) <= 2e+224)
		tmp = fma(log(y), Float64(x + -1.0), Float64(-t));
	else
		tmp = Float64(Float64(z * Float64(y * fma(y, -0.5, -1.0))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + z), $MachinePrecision], -2e+241], (-N[(y * z + t), $MachinePrecision]), If[LessEqual[N[(-1.0 + z), $MachinePrecision], 2e+224], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -2.0000000000000001e241

   1. Initial program 32.3%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 81.2

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

   5. Simplified 81.2%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. accelerator-lowering-fma.f64 81.2

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

   8. Simplified 81.2%

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

   if -2.0000000000000001e241 < (-.f64 z #s(literal 1 binary64)) < 1.99999999999999994e224

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. neg-lowering-neg.f64 96.5

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

   5. Simplified 96.5%

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

   if 1.99999999999999994e224 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 52.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 76.7

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

   5. Simplified 76.7%

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

   6. Taylor expanded in y around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 76.7

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

   8. Simplified 76.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + z \leq -2 \cdot 10^{+241}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;-1 + z \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq 10^{+49}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ x -1.0) -5e+55)
     t_1
     (if (<= (+ x -1.0) 1e+49) (- (- t) (log y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((x + -1.0) <= -5e+55) {
		tmp = t_1;
	} else if ((x + -1.0) <= 1e+49) {
		tmp = -t - log(y);
	} 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 = x * log(y)
    if ((x + (-1.0d0)) <= (-5d+55)) then
        tmp = t_1
    else if ((x + (-1.0d0)) <= 1d+49) then
        tmp = -t - log(y)
    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 = x * Math.log(y);
	double tmp;
	if ((x + -1.0) <= -5e+55) {
		tmp = t_1;
	} else if ((x + -1.0) <= 1e+49) {
		tmp = -t - Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (x + -1.0) <= -5e+55:
		tmp = t_1
	elif (x + -1.0) <= 1e+49:
		tmp = -t - math.log(y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(x + -1.0) <= -5e+55)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= 1e+49)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((x + -1.0) <= -5e+55)
		tmp = t_1;
	elseif ((x + -1.0) <= 1e+49)
		tmp = -t - log(y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -5e+55], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], 1e+49], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5.00000000000000046e55 or 9.99999999999999946e48 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 74.5

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

   5. Simplified 74.5%

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

   if -5.00000000000000046e55 < (-.f64 x #s(literal 1 binary64)) < 9.99999999999999946e48

   1. Initial program 85.2%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. neg-lowering-neg.f64 84.3

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

   7. Simplified 84.3%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

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

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

      7. log-lowering-log.f64 79.0

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

   10. Simplified 79.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x + -1 \leq 10^{+49}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq 10^{+49}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ x -1.0) -5e+55)
     t_1
     (if (<= (+ x -1.0) 1e+49)
       (- (* z (* y (fma y (fma y -0.3333333333333333 -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 + -1.0) <= -5e+55) {
		tmp = t_1;
	} else if ((x + -1.0) <= 1e+49) {
		tmp = (z * (y * fma(y, fma(y, -0.3333333333333333, -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 (Float64(x + -1.0) <= -5e+55)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= 1e+49)
		tmp = Float64(Float64(z * Float64(y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0))) - 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[N[(x + -1.0), $MachinePrecision], -5e+55], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], 1e+49], N[(N[(z * N[(y * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5.00000000000000046e55 or 9.99999999999999946e48 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 74.5

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

   5. Simplified 74.5%

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

   if -5.00000000000000046e55 < (-.f64 x #s(literal 1 binary64)) < 9.99999999999999946e48

   1. Initial program 85.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 67.5

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

   5. Simplified 67.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)} \cdot z - t \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)} \cdot z - t \]

      2. sub-neg N/A

         \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot z - t \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \cdot z - t \]

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 67.5

         \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.3333333333333333, -0.5\right)}, -1\right)\right) \cdot z - t \]

   8. Simplified 67.5%

      \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right)} \cdot z - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x + -1 \leq 10^{+49}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+64}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+20}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.2e+64)
   (- (fma y z t))
   (if (<= t 4.1e+20)
     (* (+ x -1.0) (log y))
     (- (* z (* y (fma y -0.5 -1.0))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e+64) {
		tmp = -fma(y, z, t);
	} else if (t <= 4.1e+20) {
		tmp = (x + -1.0) * log(y);
	} else {
		tmp = (z * (y * fma(y, -0.5, -1.0))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.2e+64)
		tmp = Float64(-fma(y, z, t));
	elseif (t <= 4.1e+20)
		tmp = Float64(Float64(x + -1.0) * log(y));
	else
		tmp = Float64(Float64(z * Float64(y * fma(y, -0.5, -1.0))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -8.2e+64], (-N[(y * z + t), $MachinePrecision]), If[LessEqual[t, 4.1e+20], N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.19999999999999956e64

   1. Initial program 89.8%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 93.7

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

   5. Simplified 93.7%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. accelerator-lowering-fma.f64 93.7

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

   8. Simplified 93.7%

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

   if -8.19999999999999956e64 < t < 4.1e20

   1. Initial program 89.2%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. neg-lowering-neg.f64 88.8

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

   7. Simplified 88.8%

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

   8. Taylor expanded in t around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 82.0

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

   10. Simplified 82.0%

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

   if 4.1e20 < t

   1. Initial program 94.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 77.3

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

   5. Simplified 77.3%

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

   6. Taylor expanded in y around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 77.3

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

   8. Simplified 77.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+64}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+20}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. --lowering--.f64 N/A

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

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

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

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

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. +-lowering-+.f64 99.6

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 11: 42.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-57}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+20}:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.9e-57) (- t) (if (<= t 7.2e+20) (- (* y z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.9e-57) {
		tmp = -t;
	} else if (t <= 7.2e+20) {
		tmp = -(y * z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.9d-57)) then
        tmp = -t
    else if (t <= 7.2d+20) then
        tmp = -(y * z)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.9e-57) {
		tmp = -t;
	} else if (t <= 7.2e+20) {
		tmp = -(y * z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.9e-57:
		tmp = -t
	elif t <= 7.2e+20:
		tmp = -(y * z)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.9e-57)
		tmp = Float64(-t);
	elseif (t <= 7.2e+20)
		tmp = Float64(-Float64(y * z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.9e-57)
		tmp = -t;
	elseif (t <= 7.2e+20)
		tmp = -(y * z);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.9e-57], (-t), If[LessEqual[t, 7.2e+20], (-N[(y * z), $MachinePrecision]), (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -2.90000000000000025e-57 or 7.2e20 < t

   1. Initial program 94.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 68.1

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

   5. Simplified 68.1%

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

   if -2.90000000000000025e-57 < t < 7.2e20

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z 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 16.2

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

   8. Simplified 16.2%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-57}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+20}:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.4% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

   5. neg-lowering-neg.f64 49.1

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0

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

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

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. accelerator-lowering-fma.f64 49.1

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

8. Simplified 49.1%

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

9. Final simplification 49.1%

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

Alternative 13: 46.3% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. --lowering--.f64 N/A

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

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

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

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

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. +-lowering-+.f64 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in y around inf

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

   1. distribute-rgt-out-- N/A

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

   2. *-lft-identity N/A

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

   3. *-commutative N/A

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

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

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

   5. *-lowering-*.f64 49.1

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

8. Simplified 49.1%

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

Alternative 14: 46.1% accurate, 25.1× 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 90.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

   5. neg-lowering-neg.f64 49.1

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

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

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

   5. accelerator-lowering-fma.f64 49.0

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

8. Simplified 49.0%

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

Alternative 15: 36.2% accurate, 75.3× 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 90.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 39.8

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

5. Simplified 39.8%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))