Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 16.9s

Alternatives: 14

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: 98.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	tmp = s * -log(((single(1.0) / ((single(1.0) / (single(1.0) + t_0)) + ((u / (single(1.0) + exp((single(pi) / -s)))) + (u / (single(-1.0) - t_0))))) + 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. Taylor expanded in s around 0 98.9%

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

   1. sub-neg 98.9%

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

   2. associate--l+ 98.9%

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

   3. neg-mul-1 98.9%

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

   4. distribute-neg-frac 98.9%

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

   5. metadata-eval 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((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. Final simplification 98.9%

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

Alternative 3: 25.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 24.1%

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

   1. cancel-sign-sub-inv 24.1%

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

   2. metadata-eval 24.1%

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

   3. distribute-rgt-out-- 24.1%

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

   4. metadata-eval 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + 0.25 \cdot \pi}{s}\right) \]

   5. *-commutative 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 24.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   2. sqrt-unprod 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   3. sqr-neg 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\sqrt{\color{blue}{\left(-\pi\right) \cdot \left(-\pi\right)}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   4. sqrt-unprod -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{-\pi} \cdot \sqrt{-\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   5. add-sqr-sqrt 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(-\pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   6. neg-sub0 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

7. Applied egg-rr 24.5%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

8. Taylor expanded in s around 0 24.7%

   \[\leadsto \color{blue}{s \cdot \left(-1 \cdot \left(\log \left(4 \cdot \left(0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + -1 \cdot \log s\right) + -0.25 \cdot \frac{s}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)}\right)} \]
9. Step-by-step derivation

   1. +-commutative 24.7%

      \[\leadsto s \cdot \color{blue}{\left(-0.25 \cdot \frac{s}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)} + -1 \cdot \left(\log \left(4 \cdot \left(0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + -1 \cdot \log s\right)\right)} \]

   2. mul-1-neg 24.7%

      \[\leadsto s \cdot \left(-0.25 \cdot \frac{s}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)} + \color{blue}{\left(-\left(\log \left(4 \cdot \left(0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + -1 \cdot \log s\right)\right)}\right) \]

   3. unsub-neg 24.7%

      \[\leadsto s \cdot \color{blue}{\left(-0.25 \cdot \frac{s}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)} - \left(\log \left(4 \cdot \left(0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + -1 \cdot \log s\right)\right)} \]

10. Simplified 24.7%

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

11. Taylor expanded in u around 0 24.7%

    \[\leadsto s \cdot \left(\color{blue}{-1 \cdot \frac{s}{\pi}} - \left(\log \left(\left(\pi \cdot \left(u \cdot 0.5 + 0.25\right)\right) \cdot 4\right) - \log s\right)\right) \]
12. Step-by-step derivation

    1. mul-1-neg 24.7%

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

    2. distribute-neg-frac2 24.7%

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

13. Simplified 24.7%

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

14. Final simplification 24.7%

    \[\leadsto s \cdot \left(\left(\log s - \log \left(\left(\pi \cdot \left(u \cdot 0.5 + 0.25\right)\right) \cdot 4\right)\right) - \frac{s}{\pi}\right) \]
15. Add Preprocessing

Alternative 4: 25.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 24.1%

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

   1. cancel-sign-sub-inv 24.1%

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

   2. metadata-eval 24.1%

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

   3. distribute-rgt-out-- 24.1%

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

   4. metadata-eval 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + 0.25 \cdot \pi}{s}\right) \]

   5. *-commutative 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 24.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   2. sqrt-unprod 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   3. sqr-neg 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\sqrt{\color{blue}{\left(-\pi\right) \cdot \left(-\pi\right)}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   4. sqrt-unprod -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{-\pi} \cdot \sqrt{-\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   5. add-sqr-sqrt 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(-\pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   6. neg-sub0 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

7. Applied egg-rr 24.5%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

8. Taylor expanded in s around 0 24.7%

   \[\leadsto \color{blue}{s \cdot \left(-1 \cdot \left(\log \left(4 \cdot \left(0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + -1 \cdot \log s\right) + -0.25 \cdot \frac{s}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)}\right)} \]
9. Step-by-step derivation

   1. +-commutative 24.7%

      \[\leadsto s \cdot \color{blue}{\left(-0.25 \cdot \frac{s}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)} + -1 \cdot \left(\log \left(4 \cdot \left(0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + -1 \cdot \log s\right)\right)} \]

   2. mul-1-neg 24.7%

      \[\leadsto s \cdot \left(-0.25 \cdot \frac{s}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)} + \color{blue}{\left(-\left(\log \left(4 \cdot \left(0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + -1 \cdot \log s\right)\right)}\right) \]

   3. unsub-neg 24.7%

      \[\leadsto s \cdot \color{blue}{\left(-0.25 \cdot \frac{s}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)} - \left(\log \left(4 \cdot \left(0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + -1 \cdot \log s\right)\right)} \]

10. Simplified 24.7%

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

11. Taylor expanded in u around 0 24.7%

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

    1. mul-1-neg 24.7%

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

    2. unsub-neg 24.7%

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

13. Simplified 24.7%

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

14. Final simplification 24.7%

    \[\leadsto s \cdot \left(\left(\log s - \frac{s}{\pi}\right) - \log \pi\right) \]
15. Add Preprocessing

Alternative 5: 25.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\pi}{s}\\ -2 \cdot \frac{\pi \cdot u}{t\_0} - s \cdot \log t\_0 \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ PI s)))) (- (* -2.0 (/ (* PI u) t_0)) (* s (log t_0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f + (((float) M_PI) / s);
	return (-2.0f * ((((float) M_PI) * u) / t_0)) - (s * logf(t_0));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(pi) / s))
	return Float32(Float32(Float32(-2.0) * Float32(Float32(Float32(pi) * u) / t_0)) - Float32(s * log(t_0)))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + (single(pi) / s);
	tmp = (single(-2.0) * ((single(pi) * u) / t_0)) - (s * log(t_0));
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 24.1%

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

   1. cancel-sign-sub-inv 24.1%

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

   2. metadata-eval 24.1%

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

   3. distribute-rgt-out-- 24.1%

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

   4. metadata-eval 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + 0.25 \cdot \pi}{s}\right) \]

   5. *-commutative 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 24.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   2. sqrt-unprod 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   3. sqr-neg 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\sqrt{\color{blue}{\left(-\pi\right) \cdot \left(-\pi\right)}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   4. sqrt-unprod -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{-\pi} \cdot \sqrt{-\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   5. add-sqr-sqrt 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(-\pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   6. neg-sub0 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

7. Applied egg-rr 24.5%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

8. Taylor expanded in u around 0 24.5%

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

9. Final simplification 24.5%

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

Alternative 6: 25.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\pi \cdot \left(u \cdot 0.5 + 0.25\right)}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (+ 1.0 (* 4.0 (/ (* PI (+ (* u 0.5) 0.25)) s))))))

C

float code(float u, float s) {
	return -s * logf((1.0f + (4.0f * ((((float) M_PI) * ((u * 0.5f) + 0.25f)) / s))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(1.0) + Float32(Float32(4.0) * Float32(Float32(Float32(pi) * Float32(Float32(u * Float32(0.5)) + Float32(0.25))) / s)))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + (single(4.0) * ((single(pi) * ((u * single(0.5)) + single(0.25))) / s))));
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 24.1%

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

   1. cancel-sign-sub-inv 24.1%

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

   2. metadata-eval 24.1%

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

   3. distribute-rgt-out-- 24.1%

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

   4. metadata-eval 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + 0.25 \cdot \pi}{s}\right) \]

   5. *-commutative 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 24.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   2. sqrt-unprod 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   3. sqr-neg 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\sqrt{\color{blue}{\left(-\pi\right) \cdot \left(-\pi\right)}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   4. sqrt-unprod -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{-\pi} \cdot \sqrt{-\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   5. add-sqr-sqrt 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(-\pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   6. neg-sub0 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

7. Applied egg-rr 24.5%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

8. Taylor expanded in u around 0 24.5%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\color{blue}{0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)}}{s}\right) \]
9. Step-by-step derivation

   1. +-commutative 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\color{blue}{0.5 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi}}{s}\right) \]

   2. associate-*r* 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\color{blue}{\left(0.5 \cdot u\right) \cdot \pi} + 0.25 \cdot \pi}{s}\right) \]

   3. distribute-rgt-out 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\color{blue}{\pi \cdot \left(0.5 \cdot u + 0.25\right)}}{s}\right) \]

   4. *-commutative 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\pi \cdot \left(\color{blue}{u \cdot 0.5} + 0.25\right)}{s}\right) \]

10. Simplified 24.5%

    \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\color{blue}{\pi \cdot \left(u \cdot 0.5 + 0.25\right)}}{s}\right) \]

11. Final simplification 24.5%

    \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\pi \cdot \left(u \cdot 0.5 + 0.25\right)}{s}\right) \]
12. Add Preprocessing

Alternative 7: 25.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 + 4 \cdot \left(\pi \cdot \frac{0.25}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (+ 1.0 (* 4.0 (* PI (/ 0.25 s)))))))

C

float code(float u, float s) {
	return -s * logf((1.0f + (4.0f * (((float) M_PI) * (0.25f / s)))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(1.0) + Float32(Float32(4.0) * Float32(Float32(pi) * Float32(Float32(0.25) / s))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + (single(4.0) * (single(pi) * (single(0.25) / s)))));
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 24.1%

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

   1. cancel-sign-sub-inv 24.1%

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

   2. metadata-eval 24.1%

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

   3. distribute-rgt-out-- 24.1%

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

   4. metadata-eval 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + 0.25 \cdot \pi}{s}\right) \]

   5. *-commutative 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 24.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 24.5%

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

   1. associate-*r/ 24.5%

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

   2. *-commutative 24.5%

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

   3. associate-/l* 24.5%

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

8. Simplified 24.5%

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

9. Final simplification 24.5%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \left(\pi \cdot \frac{0.25}{s}\right)\right) \]
10. Add Preprocessing

Alternative 8: 25.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(1 + 4 \cdot \left(\frac{\pi}{s} \cdot 0.25\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (- (log (+ 1.0 (* 4.0 (* (/ PI s) 0.25)))))))

C

float code(float u, float s) {
	return s * -logf((1.0f + (4.0f * ((((float) M_PI) / s) * 0.25f))));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(1.0) + Float32(Float32(4.0) * Float32(Float32(Float32(pi) / s) * Float32(0.25)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(1.0) + (single(4.0) * ((single(pi) / s) * single(0.25)))));
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 24.1%

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

   1. cancel-sign-sub-inv 24.1%

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

   2. metadata-eval 24.1%

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

   3. distribute-rgt-out-- 24.1%

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

   4. metadata-eval 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + 0.25 \cdot \pi}{s}\right) \]

   5. *-commutative 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 24.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 24.5%

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

   1. *-commutative 24.5%

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

8. Simplified 24.5%

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

9. Final simplification 24.5%

   \[\leadsto s \cdot \left(-\log \left(1 + 4 \cdot \left(\frac{\pi}{s} \cdot 0.25\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 25.1% accurate, 4.1× speedup

Math

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

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(s * Float32(-log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 24.1%

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

   1. cancel-sign-sub-inv 24.1%

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

   2. metadata-eval 24.1%

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

   3. distribute-rgt-out-- 24.1%

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

   4. metadata-eval 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + 0.25 \cdot \pi}{s}\right) \]

   5. *-commutative 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 24.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   2. sqrt-unprod 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   3. sqr-neg 24.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\sqrt{\color{blue}{\left(-\pi\right) \cdot \left(-\pi\right)}} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   4. sqrt-unprod -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(\sqrt{-\pi} \cdot \sqrt{-\pi}\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   5. add-sqr-sqrt 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(-\pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

   6. neg-sub0 24.5%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

7. Applied egg-rr 24.5%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot -0.5\right) + \pi \cdot 0.25}{s}\right) \]

8. Taylor expanded in u around 0 24.5%

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

   1. mul-1-neg 24.5%

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

   2. log1p-define 24.5%

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

   3. *-commutative 24.5%

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

   4. distribute-rgt-neg-in 24.5%

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

10. Simplified 24.5%

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

11. Final simplification 24.5%

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

Alternative 10: 11.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * ((single(pi) * (u * single(-0.25))) + (u * ((single(0.25) * (single(pi) / u)) + (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 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. metadata-eval 11.8%

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

   4. cancel-sign-sub-inv 11.8%

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

   5. associate-*r* 11.8%

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

   6. distribute-rgt-out 11.8%

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

   7. metadata-eval 11.8%

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

   8. associate-*r* 11.8%

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

7. Simplified 11.8%

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

8. Taylor expanded in u around inf 11.8%

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

9. Final simplification 11.8%

   \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot -0.25\right) + u \cdot \left(0.25 \cdot \frac{\pi}{u} + \pi \cdot -0.25\right)\right) \]
10. Add Preprocessing

Alternative 11: 11.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * ((u * ((single(pi) * (single(0.25) + (u * single(-0.25)))) / u)) + (single(pi) * (u * 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 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. metadata-eval 11.8%

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

   4. cancel-sign-sub-inv 11.8%

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

   5. associate-*r* 11.8%

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

   6. distribute-rgt-out 11.8%

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

   7. metadata-eval 11.8%

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

   8. associate-*r* 11.8%

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

7. Simplified 11.8%

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

8. Taylor expanded in u around inf 11.8%

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

9. Taylor expanded in u around 0 11.8%

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

    1. *-commutative 11.8%

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

    2. *-commutative 11.8%

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

    3. associate-*r* 11.8%

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

    4. *-commutative 11.8%

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

    5. distribute-lft-out 11.8%

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

11. Simplified 11.8%

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

12. Final simplification 11.8%

    \[\leadsto -4 \cdot \left(u \cdot \frac{\pi \cdot \left(0.25 + u \cdot -0.25\right)}{u} + \pi \cdot \left(u \cdot -0.25\right)\right) \]
13. Add Preprocessing

Alternative 12: 11.5% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.5) * (single(pi) * u)) + (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 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. distribute-rgt-out-- 11.8%

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

   4. *-commutative 11.8%

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

   5. metadata-eval 11.8%

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

   6. metadata-eval 11.8%

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

   7. *-commutative 11.8%

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

7. Simplified 11.8%

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

8. Final simplification 11.8%

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

Alternative 13: 11.5% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * (single(pi) * (single(0.25) + (u * single(-0.5))));
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 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. metadata-eval 11.8%

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

   4. cancel-sign-sub-inv 11.8%

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

   5. associate-*r* 11.8%

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

   6. distribute-rgt-out 11.8%

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

   7. metadata-eval 11.8%

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

   8. associate-*r* 11.8%

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

7. Simplified 11.8%

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

8. Taylor expanded in u around 0 11.8%

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

   1. +-commutative 11.8%

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

   2. associate-*r* 11.8%

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

   3. distribute-rgt-out 11.8%

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

10. Simplified 11.8%

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

11. Final simplification 11.8%

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

Alternative 14: 11.3% 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 11.6%

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

   1. mul-1-neg 11.6%

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

7. Simplified 11.6%

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

8. Final simplification 11.6%

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

Reproduce

herbie shell --seed 2024076 
(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))))