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

Percentage Accurate: 89.0% → 99.8%

Time: 18.8s

Alternatives: 18

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: 89.0% 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: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(y) * (x + -1.0)) + (Math.log1p(-y) / (1.0 / (z + -1.0)))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (x + -1.0)) + (math.log1p(-y) / (1.0 / (z + -1.0)))) - t

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. clear-num N/A

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

   10. flip-- N/A

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

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

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

   12. sub-neg N/A

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

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

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

   14. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ t_2 := \log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq -40000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 160:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right) - t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t))
        (t_2 (+ (* (log y) (+ x -1.0)) (* (+ z -1.0) (log (- 1.0 y))))))
   (if (<= t_2 -40000000000000.0)
     t_1
     (if (<= t_2 160.0)
       (- (fma y (- z) y) t)
       (if (<= t_2 2e+14) (- (- 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 = (log(y) * (x + -1.0)) + ((z + -1.0) * log((1.0 - y)));
	double tmp;
	if (t_2 <= -40000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 160.0) {
		tmp = fma(y, -z, y) - t;
	} else if (t_2 <= 2e+14) {
		tmp = -t - 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(log(y) * Float64(x + -1.0)) + Float64(Float64(z + -1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_2 <= -40000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 160.0)
		tmp = Float64(fma(y, Float64(-z), y) - t);
	elseif (t_2 <= 2e+14)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -40000000000000.0], t$95$1, If[LessEqual[t$95$2, 160.0], N[(N[(y * (-z) + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 2e+14], 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)))) < -4e13 or 2e14 < (+.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 96.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 93.7

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

   5. Simplified 93.7%

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

   if -4e13 < (+.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)))) < 160

   1. Initial program 59.0%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   5. Simplified 92.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 92.5

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

   8. Simplified 92.5%

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

   9. Taylor expanded in y around -inf

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

       1. associate-*r* N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-in N/A

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

       5. neg-mul-1 N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. mul-1-neg N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. mul-1-neg N/A

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

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

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

       14. mul-1-neg N/A

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

       15. neg-lowering-neg.f64 89.0

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

   11. Simplified 89.0%

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

   if 160 < (+.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)))) < 2e14

   1. Initial program 84.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 84.6

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

   5. Simplified 84.6%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

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

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

      6. log-lowering-log.f64 84.1

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

   8. Simplified 84.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right) \leq -40000000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right) \leq 160:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right) - t\\ \mathbf{elif}\;\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right) \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 160:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (* (log y) (+ x -1.0)) (* (+ z -1.0) (log (- 1.0 y))))))
   (if (<= t_2 -4e+117)
     t_1
     (if (<= t_2 160.0)
       (- (* z (* y (fma y (fma y -0.3333333333333333 -0.5) -1.0))) t)
       (if (<= t_2 2e+95) (- (- t) (log y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = (log(y) * (x + -1.0)) + ((z + -1.0) * log((1.0 - y)));
	double tmp;
	if (t_2 <= -4e+117) {
		tmp = t_1;
	} else if (t_2 <= 160.0) {
		tmp = (z * (y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0))) - t;
	} else if (t_2 <= 2e+95) {
		tmp = -t - log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(log(y) * Float64(x + -1.0)) + Float64(Float64(z + -1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_2 <= -4e+117)
		tmp = t_1;
	elseif (t_2 <= 160.0)
		tmp = Float64(Float64(z * Float64(y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0))) - t);
	elseif (t_2 <= 2e+95)
		tmp = Float64(Float64(-t) - log(y));
	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]}, Block[{t$95$2 = N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+117], t$95$1, If[LessEqual[t$95$2, 160.0], N[(N[(z * N[(y * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 2e+95], 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)))) < -4.0000000000000002e117 or 2.00000000000000004e95 < (+.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 99.3%

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

   3. Taylor expanded in 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 82.3

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

   5. Simplified 82.3%

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

   if -4.0000000000000002e117 < (+.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)))) < 160

   1. Initial program 72.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 72.1

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

   5. Simplified 72.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)} \cdot z - t \]
   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 72.1

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

   8. Simplified 72.1%

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

   if 160 < (+.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)))) < 2.00000000000000004e95

   1. Initial program 85.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right) - t} \]
   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 84.3

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

   5. Simplified 84.3%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

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

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

      6. log-lowering-log.f64 79.7

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

   8. Simplified 79.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right) \leq -4 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right) \leq 160:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t\\ \mathbf{elif}\;\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right) \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right) \cdot \left(z + -1\right), 1 - z\right), \mathsf{fma}\left(\log y, x + -1, -t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  y
  (fma y (* (fma y -0.3333333333333333 -0.5) (+ z -1.0)) (- 1.0 z))
  (fma (log y) (+ x -1.0) (- t))))

C

double code(double x, double y, double z, double t) {
	return fma(y, fma(y, (fma(y, -0.3333333333333333, -0.5) * (z + -1.0)), (1.0 - z)), fma(log(y), (x + -1.0), -t));
}

Julia

function code(x, y, z, t)
	return fma(y, fma(y, Float64(fma(y, -0.3333333333333333, -0.5) * Float64(z + -1.0)), Float64(1.0 - z)), fma(log(y), Float64(x + -1.0), Float64(-t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(y * N[(N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

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

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right) \cdot \left(z + -1\right), 1 - z\right), \mathsf{fma}\left(\log y, x + -1, -t\right)\right) \]
7. Add Preprocessing

Alternative 5: 94.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x -1.0) -2e+35)
   (- (* x (log y)) t)
   (if (<= (+ x -1.0) 20000000000.0)
     (- (fma y (- 1.0 z) (- (log y))) t)
     (fma (log y) (+ x -1.0) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + -1.0) <= -2e+35) {
		tmp = (x * log(y)) - t;
	} else if ((x + -1.0) <= 20000000000.0) {
		tmp = fma(y, (1.0 - z), -log(y)) - t;
	} else {
		tmp = fma(log(y), (x + -1.0), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -2e+35)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(x + -1.0) <= 20000000000.0)
		tmp = Float64(fma(y, Float64(1.0 - z), Float64(-log(y))) - t);
	else
		tmp = fma(log(y), Float64(x + -1.0), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + -1.0), $MachinePrecision], -2e+35], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x + -1.0), $MachinePrecision], 20000000000.0], N[(N[(y * N[(1.0 - z), $MachinePrecision] + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.9999999999999999e35

   1. Initial program 98.1%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 98.1

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

   5. Simplified 98.1%

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

   if -1.9999999999999999e35 < (-.f64 x #s(literal 1 binary64)) < 2e10

   1. Initial program 79.1%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 98.6

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

   5. Simplified 98.6%

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

   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 98.2

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

   8. Simplified 98.2%

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

   if 2e10 < (-.f64 x #s(literal 1 binary64))

   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. 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 95.2

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

   5. Simplified 95.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -z\right), -t\right)\\ \mathbf{elif}\;z + -1 \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -5e+155)
   (fma y (fma y (* z (fma y -0.3333333333333333 -0.5)) (- z)) (- t))
   (if (<= (+ z -1.0) 2e+90)
     (- (- t) (log y))
     (-
      (* z (* y (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0)))
      t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -5e+155) {
		tmp = fma(y, fma(y, (z * fma(y, -0.3333333333333333, -0.5)), -z), -t);
	} else if ((z + -1.0) <= 2e+90) {
		tmp = -t - log(y);
	} else {
		tmp = (z * (y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -5e+155)
		tmp = fma(y, fma(y, Float64(z * fma(y, -0.3333333333333333, -0.5)), Float64(-z)), Float64(-t));
	elseif (Float64(z + -1.0) <= 2e+90)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = Float64(Float64(z * Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -5e+155], N[(y * N[(y * N[(z * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[N[(z + -1.0), $MachinePrecision], 2e+90], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -4.9999999999999999e155

   1. Initial program 64.4%

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

   3. Taylor expanded in 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 69.9

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

   5. Simplified 69.9%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

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

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

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

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

      15. mul-1-neg N/A

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

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

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

      17. neg-lowering-neg.f64 69.9

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

   8. Simplified 69.9%

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

   if -4.9999999999999999e155 < (-.f64 z #s(literal 1 binary64)) < 1.99999999999999993e90

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

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

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

      6. log-lowering-log.f64 62.3

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

   8. Simplified 62.3%

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

   if 1.99999999999999993e90 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 64.3%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 52.5

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

   5. Simplified 52.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \cdot z - t \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \cdot z - t \]

      2. sub-neg N/A

         \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot z - t \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \cdot z - t \]

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

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \cdot z - t \]

      5. sub-neg N/A

         \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \cdot z - t \]

      6. metadata-eval N/A

         \[\leadsto \left(y \cdot \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \cdot z - t \]

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

         \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{4} \cdot y - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \cdot z - t \]

      8. sub-neg N/A

         \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \cdot z - t \]

      9. *-commutative N/A

         \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \cdot z - t \]

      10. metadata-eval N/A

          \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \cdot z - t \]

      11. accelerator-lowering-fma.f64 51.3

          \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \cdot z - t \]

   8. Simplified 51.3%

      \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \cdot z - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -z\right), -t\right)\\ \mathbf{elif}\;z + -1 \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+23}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{elif}\;t \leq 4200000000000:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.65e+23)
   (- (* z (* y (fma y -0.5 -1.0))) t)
   (if (<= t 4200000000000.0) (* (log y) (+ x -1.0)) (- (fma y z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.65e+23) {
		tmp = (z * (y * fma(y, -0.5, -1.0))) - t;
	} else if (t <= 4200000000000.0) {
		tmp = log(y) * (x + -1.0);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.65e+23)
		tmp = Float64(Float64(z * Float64(y * fma(y, -0.5, -1.0))) - t);
	elseif (t <= 4200000000000.0)
		tmp = Float64(log(y) * Float64(x + -1.0));
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.65e+23], N[(N[(z * N[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 4200000000000.0], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if t < -1.65000000000000015e23

   1. Initial program 84.8%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 79.3

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

   5. Simplified 79.3%

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

   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 79.3

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

   8. Simplified 79.3%

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

   if -1.65000000000000015e23 < t < 4.2e12

   1. Initial program 85.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 84.0

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

   5. Simplified 84.0%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} \]
   7. 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 82.7

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

   8. Simplified 82.7%

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

   if 4.2e12 < t

   1. Initial program 95.4%

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

   3. Taylor expanded in 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 78.8

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

   5. Simplified 78.8%

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

   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 78.8

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

   8. Simplified 78.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+23}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{elif}\;t \leq 4200000000000:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -5 \cdot 10^{+255}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -5e+255) (- (fma y z t)) (fma (log y) (+ x -1.0) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -5e+255) {
		tmp = -fma(y, z, t);
	} else {
		tmp = fma(log(y), (x + -1.0), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -5e+255)
		tmp = Float64(-fma(y, z, t));
	else
		tmp = fma(log(y), Float64(x + -1.0), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -5e+255], (-N[(y * z + t), $MachinePrecision]), N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -5.0000000000000002e255

   1. Initial program 39.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 91.3

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

   5. Simplified 91.3%

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

   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 91.3

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

   8. Simplified 91.3%

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

   if -5.0000000000000002e255 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 90.2%

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

   3. Taylor expanded in 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 88.9

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

   5. Simplified 88.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -5 \cdot 10^{+255}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. 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 99.1

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 10: 47.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (* y (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

3. 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 46.4

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

5. Simplified 46.4%

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

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \cdot z - t \]
7. Step-by-step derivation

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

      \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \cdot z - t \]

   2. sub-neg N/A

      \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot z - t \]

   3. metadata-eval N/A

      \[\leadsto \left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \cdot z - t \]

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

      \[\leadsto \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \cdot z - t \]

   5. sub-neg N/A

      \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \cdot z - t \]

   6. metadata-eval N/A

      \[\leadsto \left(y \cdot \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \cdot z - t \]

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

      \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{4} \cdot y - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \cdot z - t \]

   8. sub-neg N/A

      \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \cdot z - t \]

   9. *-commutative N/A

      \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \cdot z - t \]

   10. metadata-eval N/A

       \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \cdot z - t \]

   11. accelerator-lowering-fma.f64 46.1

       \[\leadsto \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \cdot z - t \]

8. Simplified 46.1%

   \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \cdot z - t \]

9. Final simplification 46.1%

   \[\leadsto z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t \]
10. Add Preprocessing

Alternative 11: 47.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -z\right), -t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma y (fma y (* z (fma y -0.3333333333333333 -0.5)) (- z)) (- t)))

C

double code(double x, double y, double z, double t) {
	return fma(y, fma(y, (z * fma(y, -0.3333333333333333, -0.5)), -z), -t);
}

Julia

function code(x, y, z, t)
	return fma(y, fma(y, Float64(z * fma(y, -0.3333333333333333, -0.5)), Float64(-z)), Float64(-t))
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(y * N[(z * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + (-t)), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

3. 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 46.4

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

5. Simplified 46.4%

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

6. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

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

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

   3. +-commutative N/A

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

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

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

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

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

   15. mul-1-neg N/A

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

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

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

   17. neg-lowering-neg.f64 46.1

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

8. Simplified 46.1%

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

Alternative 12: 47.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (* y (fma y (fma y -0.3333333333333333 -0.5) -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * fma(y, fma(y, -0.3333333333333333, -0.5), -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(y * N[(y * -0.3333333333333333 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

3. 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 46.4

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

5. Simplified 46.4%

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

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 46.1

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

8. Simplified 46.1%

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

9. Final simplification 46.1%

   \[\leadsto z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.3333333333333333, -0.5\right), -1\right)\right) - t \]
10. Add Preprocessing

Alternative 13: 42.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+40}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+44}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.2e+40) (- t) (if (<= t 6.8e+44) (* y (- z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.2e+40) {
		tmp = -t;
	} else if (t <= 6.8e+44) {
		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 <= (-3.2d+40)) then
        tmp = -t
    else if (t <= 6.8d+44) 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 <= -3.2e+40) {
		tmp = -t;
	} else if (t <= 6.8e+44) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.2e+40:
		tmp = -t
	elif t <= 6.8e+44:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.2e+40)
		tmp = Float64(-t);
	elseif (t <= 6.8e+44)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.2e+40)
		tmp = -t;
	elseif (t <= 6.8e+44)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.2e+40], (-t), If[LessEqual[t, 6.8e+44], N[(y * (-z)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -3.19999999999999981e40 or 6.8e44 < t

   1. Initial program 94.9%

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

   3. Taylor expanded in 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 74.1

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

   5. Simplified 74.1%

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

   if -3.19999999999999981e40 < t < 6.8e44

   1. Initial program 82.8%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 21.7

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

   5. Simplified 21.7%

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

   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 20.5

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

   8. Simplified 20.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   10. 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 18.5

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

   11. Simplified 18.5%

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

Alternative 14: 46.9% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. 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 46.4

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

5. Simplified 46.4%

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

6. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

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

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. metadata-eval N/A

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

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

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

   17. neg-lowering-neg.f64 46.0

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

8. Simplified 46.0%

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

Alternative 15: 46.9% 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 88.0%

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

3. 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 46.4

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

5. Simplified 46.4%

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

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 46.0

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

8. Simplified 46.0%

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

9. Final simplification 46.0%

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

Alternative 16: 46.8% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate--r+ N/A

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

   7. neg-sub0 N/A

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

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

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

5. Simplified 83.8%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

   19. +-lowering-+.f64 82.5

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

8. Simplified 82.5%

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

9. Taylor expanded in y around -inf

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

    1. associate-*r* N/A

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. distribute-rgt-in N/A

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

    5. neg-mul-1 N/A

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

    6. mul-1-neg N/A

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

    7. remove-double-neg N/A

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

    8. mul-1-neg N/A

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

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

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

    10. *-commutative N/A

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

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

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

    12. mul-1-neg N/A

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

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

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

    14. mul-1-neg N/A

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

    15. neg-lowering-neg.f64 45.7

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

11. Simplified 45.7%

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

Alternative 17: 46.6% 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 88.0%

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

3. 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 46.4

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

5. Simplified 46.4%

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

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 45.6

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

8. Simplified 45.6%

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

Alternative 18: 36.0% 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 88.0%

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

3. 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 34.2

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

5. Simplified 34.2%

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

Reproduce

herbie shell --seed 2024204 
(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))