Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.6s

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: 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, 1.3× 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(Float32(pi) / 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 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 37.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(2.0)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(Float32(pi) * log(Float32(exp(1)))) / s)))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(1.0) / ((u / single(2.0)) + ((single(1.0) - u) / (single(1.0) + exp(((single(pi) * log(single(2.71828182845904523536))) / s))))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 37.3%

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

   1. *-un-lft-identity 37.3%

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

   2. exp-prod 37.3%

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

7. Applied egg-rr 37.3%

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

   1. exp-1-e 37.3%

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

9. Simplified 37.3%

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

10. Taylor expanded in s around 0 37.3%

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

11. Final simplification 37.3%

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

Alternative 3: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(2.0)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(pi) / s)))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(1.0) / ((u / single(2.0)) + ((single(1.0) - u) / (single(1.0) + (single(2.71828182845904523536) ^ (single(pi) / s))))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 37.3%

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

   1. *-un-lft-identity 37.3%

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

   2. exp-prod 37.3%

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

7. Applied egg-rr 37.3%

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

   1. exp-1-e 37.3%

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

9. Simplified 37.3%

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

10. Final simplification 37.3%

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

Alternative 4: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 37.3%

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

6. Final simplification 37.3%

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

Alternative 5: 36.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 37.3%

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

   1. *-un-lft-identity 37.3%

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

   2. exp-prod 37.3%

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

7. Applied egg-rr 37.3%

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

   1. exp-1-e 37.3%

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

9. Simplified 37.3%

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

10. Taylor expanded in s around inf 35.5%

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

    1. log-E 35.5%

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

    2. associate-/l* 35.5%

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

12. Simplified 35.5%

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

13. Final simplification 35.5%

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

Alternative 6: 36.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u / 2.0f) + ((1.0f - u) / (1.0f + (1.0f + (((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(2.0)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(pi) / s))))))))))
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 37.3%

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

6. Taylor expanded in s around inf 35.5%

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

7. Final simplification 35.5%

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

Alternative 7: 11.5% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 12.5%

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

   1. associate--r+ 12.5%

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

   2. cancel-sign-sub-inv 12.5%

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

   3. distribute-rgt-out-- 12.5%

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

   4. metadata-eval 12.5%

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

   5. *-commutative 12.5%

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

   6. metadata-eval 12.5%

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

   7. *-commutative 12.5%

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

7. Simplified 12.5%

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

8. Taylor expanded in u around inf 12.5%

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

   1. +-commutative 12.5%

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

   2. *-commutative 12.5%

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

   3. fma-define 12.5%

      \[\leadsto 4 \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, -0.25 \cdot \frac{\pi}{u}\right)}\right) \]

   4. associate-*r/ 12.5%

      \[\leadsto 4 \cdot \left(u \cdot \mathsf{fma}\left(\pi, 0.5, \color{blue}{\frac{-0.25 \cdot \pi}{u}}\right)\right) \]

   5. *-commutative 12.5%

      \[\leadsto 4 \cdot \left(u \cdot \mathsf{fma}\left(\pi, 0.5, \frac{\color{blue}{\pi \cdot -0.25}}{u}\right)\right) \]

   6. associate-/l* 12.5%

      \[\leadsto 4 \cdot \left(u \cdot \mathsf{fma}\left(\pi, 0.5, \color{blue}{\pi \cdot \frac{-0.25}{u}}\right)\right) \]

10. Simplified 12.5%

    \[\leadsto 4 \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(\pi, 0.5, \pi \cdot \frac{-0.25}{u}\right)\right)} \]

11. Taylor expanded in u around inf 12.5%

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

    1. associate-*r/ 12.5%

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

    2. *-commutative 12.5%

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

    3. *-commutative 12.5%

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

    4. associate-/l* 12.5%

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

    5. distribute-lft-out 12.5%

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

13. Simplified 12.5%

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

14. Final simplification 12.5%

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

Alternative 8: 11.5% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 12.5%

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

   1. associate--r+ 12.5%

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

   2. cancel-sign-sub-inv 12.5%

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

   3. distribute-rgt-out-- 12.5%

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

   4. metadata-eval 12.5%

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

   5. *-commutative 12.5%

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

   6. metadata-eval 12.5%

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

   7. *-commutative 12.5%

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

7. Simplified 12.5%

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

   1. *-commutative 12.5%

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

   2. metadata-eval 12.5%

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

   3. distribute-rgt-out-- 12.5%

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

   4. fma-neg 12.5%

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

   5. *-commutative 12.5%

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

   6. associate-*r* 12.5%

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

   7. *-commutative 12.5%

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

   8. *-commutative 12.5%

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

   9. metadata-eval 12.5%

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

   10. cancel-sign-sub-inv 12.5%

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

   11. *-commutative 12.5%

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

   12. associate-*r* 12.5%

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

   13. *-commutative 12.5%

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

   14. fma-neg 12.5%

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

   15. associate--r+ 12.5%

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

   16. associate-*r* 12.5%

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

9. Applied egg-rr 12.5%

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

10. Taylor expanded in u around inf 12.5%

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

    1. associate-*r/ 12.5%

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

    2. metadata-eval 12.5%

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

12. Simplified 12.5%

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

13. Final simplification 12.5%

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

Alternative 9: 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 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 12.5%

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

   1. associate--r+ 12.5%

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

   2. cancel-sign-sub-inv 12.5%

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

   3. distribute-rgt-out-- 12.5%

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

   4. metadata-eval 12.5%

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

   5. *-commutative 12.5%

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

   6. metadata-eval 12.5%

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

   7. *-commutative 12.5%

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

7. Simplified 12.5%

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

   1. associate-*l* 12.5%

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

   2. distribute-lft-out 12.5%

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

9. Applied egg-rr 12.5%

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

10. Final simplification 12.5%

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

Alternative 10: 11.3% accurate, 86.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 37.3%

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

   1. *-un-lft-identity 37.3%

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

   2. exp-prod 37.3%

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

7. Applied egg-rr 37.3%

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

   1. exp-1-e 37.3%

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

9. Simplified 37.3%

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

10. Taylor expanded in s around inf 12.1%

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

12. Simplified 12.1%

    \[\leadsto \color{blue}{\pi \cdot \left(u - 1\right)} \]

13. Final simplification 12.1%

    \[\leadsto \pi \cdot \left(u + -1\right) \]
14. Add Preprocessing

Alternative 11: 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 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u around 0 12.0%

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

   1. neg-mul-1 12.0%

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

7. Simplified 12.0%

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

8. Final simplification 12.0%

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

Alternative 12: 4.7% accurate, 433.0× 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(pi)
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 37.3%

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

6. Taylor expanded in u around 0 12.0%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 7.3%

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

   3. sqr-neg 7.3%

      \[\leadsto \sqrt{\color{blue}{s \cdot s}} \cdot \frac{\pi}{s} \]

   4. sqrt-unprod 4.6%

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

   5. add-sqr-sqrt 4.6%

      \[\leadsto \color{blue}{s} \cdot \frac{\pi}{s} \]

   6. clear-num 4.6%

      \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{s}{\pi}}} \]

   7. un-div-inv 4.7%

      \[\leadsto \color{blue}{\frac{s}{\frac{s}{\pi}}} \]

8. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{\frac{s}{\frac{s}{\pi}}} \]
9. Step-by-step derivation

   1. associate-/r/ 4.7%

      \[\leadsto \color{blue}{\frac{s}{s} \cdot \pi} \]

   2. *-inverses 4.7%

      \[\leadsto \color{blue}{1} \cdot \pi \]

   3. *-lft-identity 4.7%

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

10. Simplified 4.7%

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

11. Final simplification 4.7%

    \[\leadsto \pi \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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))))