Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 17.6s

Alternatives: 12

Speedup: N/A×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Alternative 1: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+ (/ u (+ 1.0 (exp (/ PI (- s))))) (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))))
    -1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))))) + -1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) + Float32(-1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))))) + single(-1.0)));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Final simplification 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
6. Add Preprocessing

Alternative 2: 25.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - e^{\log \left(s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (- (* u (/ (* PI 2.0) (+ 1.0 (/ PI s)))) (exp (log (* s (log1p (/ PI s)))))))

C

float code(float u, float s) {
	return (u * ((((float) M_PI) * 2.0f) / (1.0f + (((float) M_PI) / s)))) - expf(logf((s * log1pf((((float) M_PI) / s)))));
}

Julia

function code(u, s)
	return Float32(Float32(u * Float32(Float32(Float32(pi) * Float32(2.0)) / Float32(Float32(1.0) + Float32(Float32(pi) / s)))) - exp(log(Float32(s * log1p(Float32(Float32(pi) / s))))))
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}}\right) \]

   2. +-commutative 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s} + 1\right)} \]

   3. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}} + 1\right) \]

   4. fma-define 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)} \]

7. Simplified 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right) + 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}}} \]
9. Step-by-step derivation

   1. +-commutative 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + -1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   2. mul-1-neg 25.1%

      \[\leadsto 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + \color{blue}{\left(-s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   3. unsub-neg 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} - s \cdot \log \left(1 + \frac{\pi}{s}\right)} \]

   4. *-commutative 25.1%

      \[\leadsto \color{blue}{\frac{u \cdot \pi}{1 + \frac{\pi}{s}} \cdot 2} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   5. associate-/l* 25.1%

      \[\leadsto \color{blue}{\left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right)} \cdot 2 - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   6. associate-*r* 25.1%

      \[\leadsto \color{blue}{u \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} \cdot 2\right)} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   7. associate-*l/ 25.1%

      \[\leadsto u \cdot \color{blue}{\frac{\pi \cdot 2}{1 + \frac{\pi}{s}}} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   8. log1p-define 25.1%

      \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Simplified 25.1%

    \[\leadsto \color{blue}{u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]
11. Step-by-step derivation

    1. add-exp-log 25.1%

       \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - \color{blue}{e^{\log \left(s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}} \]

12. Applied egg-rr 25.1%

    \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - \color{blue}{e^{\log \left(s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}} \]

13. Final simplification 25.1%

    \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - e^{\log \left(s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]
14. Add Preprocessing

Alternative 3: 25.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (- (* u (/ (* PI 2.0) (+ 1.0 (/ PI s)))) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return (u * ((((float) M_PI) * 2.0f) / (1.0f + (((float) M_PI) / s)))) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(u * Float32(Float32(Float32(pi) * Float32(2.0)) / Float32(Float32(1.0) + Float32(Float32(pi) / s)))) - Float32(s * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}}\right) \]

   2. +-commutative 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s} + 1\right)} \]

   3. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}} + 1\right) \]

   4. fma-define 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)} \]

7. Simplified 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right) + 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}}} \]
9. Step-by-step derivation

   1. +-commutative 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + -1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   2. mul-1-neg 25.1%

      \[\leadsto 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + \color{blue}{\left(-s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   3. unsub-neg 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} - s \cdot \log \left(1 + \frac{\pi}{s}\right)} \]

   4. *-commutative 25.1%

      \[\leadsto \color{blue}{\frac{u \cdot \pi}{1 + \frac{\pi}{s}} \cdot 2} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   5. associate-/l* 25.1%

      \[\leadsto \color{blue}{\left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right)} \cdot 2 - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   6. associate-*r* 25.1%

      \[\leadsto \color{blue}{u \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} \cdot 2\right)} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   7. associate-*l/ 25.1%

      \[\leadsto u \cdot \color{blue}{\frac{\pi \cdot 2}{1 + \frac{\pi}{s}}} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   8. log1p-define 25.1%

      \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Simplified 25.1%

    \[\leadsto \color{blue}{u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

11. Final simplification 25.1%

    \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
12. Add Preprocessing

Alternative 4: 25.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 2 - \frac{s \cdot \log \left(1 + \frac{\pi}{s}\right)}{u}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* u (- (* s 2.0) (/ (* s (log (+ 1.0 (/ PI s)))) u))))

C

float code(float u, float s) {
	return u * ((s * 2.0f) - ((s * logf((1.0f + (((float) M_PI) / s)))) / u));
}

Julia

function code(u, s)
	return Float32(u * Float32(Float32(s * Float32(2.0)) - Float32(Float32(s * log(Float32(Float32(1.0) + Float32(Float32(pi) / s)))) / u)))
end

MATLAB

function tmp = code(u, s)
	tmp = u * ((s * single(2.0)) - ((s * log((single(1.0) + (single(pi) / s)))) / u));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}}\right) \]

   2. +-commutative 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s} + 1\right)} \]

   3. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}} + 1\right) \]

   4. fma-define 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)} \]

7. Simplified 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right) + 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}}} \]
9. Step-by-step derivation

   1. +-commutative 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + -1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   2. mul-1-neg 25.1%

      \[\leadsto 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + \color{blue}{\left(-s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   3. unsub-neg 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} - s \cdot \log \left(1 + \frac{\pi}{s}\right)} \]

   4. *-commutative 25.1%

      \[\leadsto \color{blue}{\frac{u \cdot \pi}{1 + \frac{\pi}{s}} \cdot 2} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   5. associate-/l* 25.1%

      \[\leadsto \color{blue}{\left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right)} \cdot 2 - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   6. associate-*r* 25.1%

      \[\leadsto \color{blue}{u \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} \cdot 2\right)} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   7. associate-*l/ 25.1%

      \[\leadsto u \cdot \color{blue}{\frac{\pi \cdot 2}{1 + \frac{\pi}{s}}} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   8. log1p-define 25.1%

      \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Simplified 25.1%

    \[\leadsto \color{blue}{u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

11. Taylor expanded in s around 0 25.1%

    \[\leadsto u \cdot \color{blue}{\left(2 \cdot s\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

12. Taylor expanded in u around inf 25.1%

    \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{s \cdot \log \left(1 + \frac{\pi}{s}\right)}{u} + 2 \cdot s\right)} \]

13. Final simplification 25.1%

    \[\leadsto u \cdot \left(s \cdot 2 - \frac{s \cdot \log \left(1 + \frac{\pi}{s}\right)}{u}\right) \]
14. Add Preprocessing

Alternative 5: 25.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 2\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- (* u (* s 2.0)) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return (u * (s * 2.0f)) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(u * Float32(s * Float32(2.0))) - Float32(s * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}}\right) \]

   2. +-commutative 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s} + 1\right)} \]

   3. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}} + 1\right) \]

   4. fma-define 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)} \]

7. Simplified 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right) + 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}}} \]
9. Step-by-step derivation

   1. +-commutative 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + -1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   2. mul-1-neg 25.1%

      \[\leadsto 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + \color{blue}{\left(-s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   3. unsub-neg 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} - s \cdot \log \left(1 + \frac{\pi}{s}\right)} \]

   4. *-commutative 25.1%

      \[\leadsto \color{blue}{\frac{u \cdot \pi}{1 + \frac{\pi}{s}} \cdot 2} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   5. associate-/l* 25.1%

      \[\leadsto \color{blue}{\left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right)} \cdot 2 - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   6. associate-*r* 25.1%

      \[\leadsto \color{blue}{u \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} \cdot 2\right)} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   7. associate-*l/ 25.1%

      \[\leadsto u \cdot \color{blue}{\frac{\pi \cdot 2}{1 + \frac{\pi}{s}}} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   8. log1p-define 25.1%

      \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Simplified 25.1%

    \[\leadsto \color{blue}{u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

11. Taylor expanded in s around 0 25.1%

    \[\leadsto u \cdot \color{blue}{\left(2 \cdot s\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

12. Final simplification 25.1%

    \[\leadsto u \cdot \left(s \cdot 2\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
13. Add Preprocessing

Alternative 6: 25.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot 2 - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (- (* u 2.0) (log1p (/ PI s)))))

C

float code(float u, float s) {
	return s * ((u * 2.0f) - log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(u * Float32(2.0)) - log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}}\right) \]

   2. +-commutative 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s} + 1\right)} \]

   3. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}} + 1\right) \]

   4. fma-define 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)} \]

7. Simplified 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right) + 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}}} \]
9. Step-by-step derivation

   1. +-commutative 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + -1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   2. mul-1-neg 25.1%

      \[\leadsto 2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} + \color{blue}{\left(-s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]

   3. unsub-neg 25.1%

      \[\leadsto \color{blue}{2 \cdot \frac{u \cdot \pi}{1 + \frac{\pi}{s}} - s \cdot \log \left(1 + \frac{\pi}{s}\right)} \]

   4. *-commutative 25.1%

      \[\leadsto \color{blue}{\frac{u \cdot \pi}{1 + \frac{\pi}{s}} \cdot 2} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   5. associate-/l* 25.1%

      \[\leadsto \color{blue}{\left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right)} \cdot 2 - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   6. associate-*r* 25.1%

      \[\leadsto \color{blue}{u \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} \cdot 2\right)} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   7. associate-*l/ 25.1%

      \[\leadsto u \cdot \color{blue}{\frac{\pi \cdot 2}{1 + \frac{\pi}{s}}} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   8. log1p-define 25.1%

      \[\leadsto u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Simplified 25.1%

    \[\leadsto \color{blue}{u \cdot \frac{\pi \cdot 2}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

11. Taylor expanded in s around 0 25.1%

    \[\leadsto u \cdot \color{blue}{\left(2 \cdot s\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
12. Step-by-step derivation

    1. associate-*r* 25.1%

       \[\leadsto \color{blue}{\left(u \cdot 2\right) \cdot s} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

    2. *-commutative 25.1%

       \[\leadsto \left(u \cdot 2\right) \cdot s - \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot s} \]

    3. distribute-rgt-out-- 25.1%

       \[\leadsto \color{blue}{s \cdot \left(u \cdot 2 - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]

13. Applied egg-rr 25.1%

    \[\leadsto \color{blue}{s \cdot \left(u \cdot 2 - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]

14. Final simplification 25.1%

    \[\leadsto s \cdot \left(u \cdot 2 - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]
15. Add Preprocessing

Alternative 7: 25.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}}\right) \]

   2. +-commutative 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s} + 1\right)} \]

   3. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}} + 1\right) \]

   4. fma-define 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)} \]

7. Simplified 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]
9. Step-by-step derivation

   1. mul-1-neg 25.1%

      \[\leadsto \color{blue}{-s \cdot \log \left(1 + \frac{\pi}{s}\right)} \]

   2. log1p-define 25.1%

      \[\leadsto -s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

   3. *-commutative 25.1%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot s} \]

   4. distribute-rgt-neg-in 25.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]

10. Simplified 25.1%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]

11. Final simplification 25.1%

    \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
12. Add Preprocessing

Alternative 8: 12.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{{s}^{2}}{-\pi} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (/ (pow s 2.0) (- PI)))

C

float code(float u, float s) {
	return powf(s, 2.0f) / -((float) M_PI);
}

Julia

function code(u, s)
	return Float32((s ^ Float32(2.0)) / Float32(-Float32(pi)))
end

MATLAB

function tmp = code(u, s)
	tmp = (s ^ single(2.0)) / -single(pi);
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}}\right) \]

   2. +-commutative 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s} + 1\right)} \]

   3. associate-*r/ 24.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}} + 1\right) \]

   4. fma-define 24.8%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)} \]

7. Simplified 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]
9. Step-by-step derivation

   1. mul-1-neg 25.1%

      \[\leadsto \color{blue}{-s \cdot \log \left(1 + \frac{\pi}{s}\right)} \]

   2. log1p-define 25.1%

      \[\leadsto -s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

   3. *-commutative 25.1%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot s} \]

   4. distribute-rgt-neg-in 25.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]

10. Simplified 25.1%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]

11. Taylor expanded in s around 0 25.1%

    \[\leadsto \color{blue}{\left(\log \pi + \left(-1 \cdot \log s + \frac{s}{\pi}\right)\right)} \cdot \left(-s\right) \]
12. Step-by-step derivation

    1. +-commutative 25.1%

       \[\leadsto \color{blue}{\left(\left(-1 \cdot \log s + \frac{s}{\pi}\right) + \log \pi\right)} \cdot \left(-s\right) \]

    2. +-commutative 25.1%

       \[\leadsto \left(\color{blue}{\left(\frac{s}{\pi} + -1 \cdot \log s\right)} + \log \pi\right) \cdot \left(-s\right) \]

    3. associate-+l+ 25.1%

       \[\leadsto \color{blue}{\left(\frac{s}{\pi} + \left(-1 \cdot \log s + \log \pi\right)\right)} \cdot \left(-s\right) \]

    4. +-commutative 25.1%

       \[\leadsto \left(\frac{s}{\pi} + \color{blue}{\left(\log \pi + -1 \cdot \log s\right)}\right) \cdot \left(-s\right) \]

    5. mul-1-neg 25.1%

       \[\leadsto \left(\frac{s}{\pi} + \left(\log \pi + \color{blue}{\left(-\log s\right)}\right)\right) \cdot \left(-s\right) \]

    6. unsub-neg 25.1%

       \[\leadsto \left(\frac{s}{\pi} + \color{blue}{\left(\log \pi - \log s\right)}\right) \cdot \left(-s\right) \]

13. Simplified 25.1%

    \[\leadsto \color{blue}{\left(\frac{s}{\pi} + \left(\log \pi - \log s\right)\right)} \cdot \left(-s\right) \]

14. Taylor expanded in s around inf 12.6%

    \[\leadsto \color{blue}{-1 \cdot \frac{{s}^{2}}{\pi}} \]
15. Step-by-step derivation

    1. mul-1-neg 12.6%

       \[\leadsto \color{blue}{-\frac{{s}^{2}}{\pi}} \]

    2. distribute-neg-frac2 12.6%

       \[\leadsto \color{blue}{\frac{{s}^{2}}{-\pi}} \]

16. Simplified 12.6%

    \[\leadsto \color{blue}{\frac{{s}^{2}}{-\pi}} \]

17. Final simplification 12.6%

    \[\leadsto \frac{{s}^{2}}{-\pi} \]
18. Add Preprocessing

Alternative 9: 11.6% accurate, 39.4× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(\left(u \cdot \pi\right) \cdot 0.5 + \pi \cdot -0.25\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* 4.0 (+ (* (* u PI) 0.5) (* PI -0.25))))

C

float code(float u, float s) {
	return 4.0f * (((u * ((float) M_PI)) * 0.5f) + (((float) M_PI) * -0.25f));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(u * Float32(pi)) * Float32(0.5)) + Float32(Float32(pi) * Float32(-0.25))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (((u * single(pi)) * single(0.5)) + (single(pi) * single(-0.25)));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 12.0%

   \[\leadsto \color{blue}{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. associate--r+ 12.0%

      \[\leadsto 4 \cdot \color{blue}{\left(\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) - 0.25 \cdot \pi\right)} \]

   2. cancel-sign-sub-inv 12.0%

      \[\leadsto 4 \cdot \color{blue}{\left(\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) + \left(-0.25\right) \cdot \pi\right)} \]

   3. distribute-rgt-out-- 12.0%

      \[\leadsto 4 \cdot \left(\color{blue}{\left(u \cdot \pi\right) \cdot \left(0.25 - -0.25\right)} + \left(-0.25\right) \cdot \pi\right) \]

   4. *-commutative 12.0%

      \[\leadsto 4 \cdot \left(\color{blue}{\left(\pi \cdot u\right)} \cdot \left(0.25 - -0.25\right) + \left(-0.25\right) \cdot \pi\right) \]

   5. metadata-eval 12.0%

      \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi\right) \]

   6. metadata-eval 12.0%

      \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi\right) \]

   7. *-commutative 12.0%

      \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}\right) \]

7. Simplified 12.0%

   \[\leadsto \color{blue}{4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25\right)} \]

8. Final simplification 12.0%

   \[\leadsto 4 \cdot \left(\left(u \cdot \pi\right) \cdot 0.5 + \pi \cdot -0.25\right) \]
9. Add Preprocessing

Alternative 10: 11.6% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(\pi \cdot \left(u \cdot -0.5 + 0.25\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* -4.0 (* PI (+ (* u -0.5) 0.25))))

C

float code(float u, float s) {
	return -4.0f * (((float) M_PI) * ((u * -0.5f) + 0.25f));
}

Julia

function code(u, s)
	return Float32(Float32(-4.0) * Float32(Float32(pi) * Float32(Float32(u * Float32(-0.5)) + Float32(0.25))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * (single(pi) * ((u * single(-0.5)) + single(0.25)));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf 12.0%

   \[\leadsto \color{blue}{-4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate--r+ 12.0%

      \[\leadsto -4 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \pi\right) - 0.25 \cdot \left(u \cdot \pi\right)\right)} \]

   2. cancel-sign-sub-inv 12.0%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(-0.25 \cdot \left(u \cdot \pi\right) + \left(--0.25\right) \cdot \pi\right)} - 0.25 \cdot \left(u \cdot \pi\right)\right) \]

   3. metadata-eval 12.0%

      \[\leadsto -4 \cdot \left(\left(-0.25 \cdot \left(u \cdot \pi\right) + \color{blue}{0.25} \cdot \pi\right) - 0.25 \cdot \left(u \cdot \pi\right)\right) \]

   4. cancel-sign-sub-inv 12.0%

      \[\leadsto -4 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right)} \]

   5. associate-*r* 12.0%

      \[\leadsto -4 \cdot \left(\left(\color{blue}{\left(-0.25 \cdot u\right) \cdot \pi} + 0.25 \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   6. distribute-rgt-out 12.0%

      \[\leadsto -4 \cdot \left(\color{blue}{\pi \cdot \left(-0.25 \cdot u + 0.25\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   7. *-commutative 12.0%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(\color{blue}{u \cdot -0.25} + 0.25\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   8. metadata-eval 12.0%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \color{blue}{-0.25} \cdot \left(u \cdot \pi\right)\right) \]

   9. *-commutative 12.0%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \color{blue}{\left(u \cdot \pi\right) \cdot -0.25}\right) \]

   10. associate-*l* 12.0%

       \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \color{blue}{u \cdot \left(\pi \cdot -0.25\right)}\right) \]

7. Simplified 12.0%

   \[\leadsto \color{blue}{-4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)} \]

8. Taylor expanded in u around 0 12.0%

   \[\leadsto -4 \cdot \color{blue}{\left(-0.5 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)} \]
9. Step-by-step derivation

   1. associate-*r* 12.0%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(-0.5 \cdot u\right) \cdot \pi} + 0.25 \cdot \pi\right) \]

   2. distribute-rgt-out 12.0%

      \[\leadsto -4 \cdot \color{blue}{\left(\pi \cdot \left(-0.5 \cdot u + 0.25\right)\right)} \]

   3. *-commutative 12.0%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(\color{blue}{u \cdot -0.5} + 0.25\right)\right) \]

10. Simplified 12.0%

    \[\leadsto -4 \cdot \color{blue}{\left(\pi \cdot \left(u \cdot -0.5 + 0.25\right)\right)} \]

11. Final simplification 12.0%

    \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot -0.5 + 0.25\right)\right) \]
12. Add Preprocessing

Alternative 11: 11.3% accurate, 72.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{\pi}{-s} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (/ PI (- s))))

C

float code(float u, float s) {
	return s * (((float) M_PI) / -s);
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(pi) / Float32(-s)))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (single(pi) / -s);
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in u around 0 6.2%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(e^{\frac{\pi}{s}}\right)} \]

6. Taylor expanded in s around 0 11.8%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\pi}{s}} \]

7. Final simplification 11.8%

   \[\leadsto s \cdot \frac{\pi}{-s} \]
8. Add Preprocessing

Alternative 12: 11.4% accurate, 216.5× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 (- PI))

C

float code(float u, float s) {
	return -((float) M_PI);
}

Julia

function code(u, s)
	return Float32(-Float32(pi))
end

MATLAB

function tmp = code(u, s)
	tmp = -single(pi);
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in u around 0 11.8%

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

   1. neg-mul-1 11.8%

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

7. Simplified 11.8%

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

8. Final simplification 11.8%

   \[\leadsto -\pi \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))