Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.1s

Alternatives: 14

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (pow E (/ 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 + powf(((float) M_E), (((float) M_PI) / s)))))) + -1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + (Float32(exp(1)) ^ 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) + (single(2.71828182845904523536) ^ (single(pi) / s)))))) + single(-1.0)));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

   1. *-un-lft-identity 98.7%

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

   2. exp-prod 98.7%

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

   3. exp-1-e 98.7%

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

5. Applied egg-rr 98.7%

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

6. Final simplification 98.7%

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

Alternative 2: 98.9% accurate, 1.4× speedup

Math

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

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(Float32(-s) * 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 98.7%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 3: 86.3% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (+ 1.0 (/ 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 + (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(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + 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(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + (single(1.0) + (single(pi) / s))))))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 84.7%

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

   1. +-commutative 36.0%

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

6. Simplified 84.7%

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

7. Final simplification 84.7%

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

Alternative 4: 37.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / (((1.0f - u) / (1.0f + powf(((float) M_E), (((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) + (Float32(exp(1)) ^ 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) + (single(2.71828182845904523536) ^ (single(pi) / s)))) + (u / single(2.0))))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{0 - s \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
6. Step-by-step derivation

   1. neg-sub0 6.8%

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

   2. distribute-rgt-neg-in 6.8%

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

   3. +-commutative 6.8%

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

7. Simplified 6.8%

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

8. Taylor expanded in s around inf 38.2%

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

   1. *-un-lft-identity 98.7%

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

   2. exp-prod 98.7%

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

   3. exp-1-e 98.7%

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

10. Applied egg-rr 38.2%

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

11. Final simplification 38.2%

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

Alternative 5: 37.7% accurate, 2.3× speedup

Math

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

FPCore

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

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 * 0.5f)))));
}

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(0.5))))))))
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(0.5))))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{0 - s \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
6. Step-by-step derivation

   1. neg-sub0 6.8%

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

   2. distribute-rgt-neg-in 6.8%

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

   3. +-commutative 6.8%

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

7. Simplified 6.8%

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

8. Taylor expanded in s around inf 38.2%

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

   1. distribute-rgt-neg-out 38.2%

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

   2. div-inv 38.2%

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

   3. metadata-eval 38.2%

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

   4. metadata-eval 38.2%

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

10. Applied egg-rr 38.2%

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

11. Final simplification 38.2%

    \[\leadsto s \cdot \left(-\log \left(-1 + \frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + u \cdot 0.5}\right)\right) \]

Alternative 6: 37.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(Float32(1.0) / u) / Float32(Float32(0.5) + Float32(Float32(-1.0) / 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(0.5) + (single(-1.0) / (single(1.0) + exp((single(pi) / s))))))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{0 - s \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
6. Step-by-step derivation

   1. neg-sub0 6.8%

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

   2. distribute-rgt-neg-in 6.8%

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

   3. +-commutative 6.8%

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

7. Simplified 6.8%

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

8. Taylor expanded in s around inf 38.2%

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

   1. *-un-lft-identity 98.7%

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

   2. exp-prod 98.7%

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

   3. exp-1-e 98.7%

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

10. Applied egg-rr 38.2%

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

11. Taylor expanded in u around inf 37.1%

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

    1. sub-neg 37.1%

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

    2. associate-/r* 37.1%

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

    3. log-E 37.1%

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

    4. *-commutative 37.1%

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

    5. *-lft-identity 37.1%

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

    6. metadata-eval 37.1%

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

13. Simplified 37.1%

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

14. Final simplification 37.1%

    \[\leadsto s \cdot \left(-\log \left(-1 + \frac{\frac{1}{u}}{0.5 + \frac{-1}{1 + e^{\frac{\pi}{s}}}}\right)\right) \]

Alternative 7: 36.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{0 - s \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
6. Step-by-step derivation

   1. neg-sub0 6.8%

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

   2. distribute-rgt-neg-in 6.8%

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

   3. +-commutative 6.8%

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

7. Simplified 6.8%

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

8. Taylor expanded in s around inf 38.2%

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

9. Taylor expanded in s around inf 36.0%

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

    1. +-commutative 36.0%

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

11. Simplified 36.0%

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

12. Final simplification 36.0%

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

Alternative 8: 25.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf((1.0f + (4.0f * ((((float) M_PI) * ((u * -0.25f) + 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(-0.25)) + Float32(0.25))) / s)))))
end

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{0 - s \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
6. Step-by-step derivation

   1. neg-sub0 6.8%

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

   2. distribute-rgt-neg-in 6.8%

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

   3. +-commutative 6.8%

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

7. Simplified 6.8%

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

8. Taylor expanded in s around inf 38.2%

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

   1. *-un-lft-identity 98.7%

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

   2. exp-prod 98.7%

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

   3. exp-1-e 98.7%

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

10. Applied egg-rr 38.2%

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

11. Taylor expanded in s around -inf 25.4%

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

    1. cancel-sign-sub-inv 25.4%

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

    2. metadata-eval 25.4%

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

    3. log-E 25.4%

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

    4. associate-*r* 25.4%

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

    5. *-rgt-identity 25.4%

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

    6. associate-*r* 25.4%

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

    7. log-E 25.4%

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

    8. associate-*r* 25.4%

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

    9. *-rgt-identity 25.4%

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

    10. distribute-rgt-out 25.4%

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

13. Simplified 25.4%

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

14. Final simplification 25.4%

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

Alternative 9: 16.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-4 \cdot \left(u \cdot \left(--0.5\right) - \frac{u}{2 - \frac{\pi}{s}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (* -4.0 (- (* u (- -0.5)) (/ u (- 2.0 (/ PI s)))))))

C

float code(float u, float s) {
	return s * (-4.0f * ((u * -(-0.5f)) - (u / (2.0f - (((float) M_PI) / s)))));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(-4.0) * Float32(Float32(u * Float32(-Float32(-0.5))) - Float32(u / Float32(Float32(2.0) - Float32(Float32(pi) / s))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (single(-4.0) * ((u * -single(-0.5)) - (u / (single(2.0) - (single(pi) / s)))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 8.8%

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

5. Taylor expanded in s around inf 15.4%

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

   1. mul-1-neg 15.4%

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

   2. unsub-neg 15.4%

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

7. Simplified 15.4%

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

8. Taylor expanded in u around 0 16.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \left(u \cdot \left(\frac{1}{2 - \frac{\pi}{s}} - 0.5\right)\right)\right)} \]
9. Step-by-step derivation

   1. sub-neg 16.1%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \left(u \cdot \color{blue}{\left(\frac{1}{2 - \frac{\pi}{s}} + \left(-0.5\right)\right)}\right)\right) \]

   2. metadata-eval 16.1%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \left(u \cdot \left(\frac{1}{2 - \frac{\pi}{s}} + \color{blue}{-0.5}\right)\right)\right) \]

   3. distribute-lft-in 16.1%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \color{blue}{\left(u \cdot \frac{1}{2 - \frac{\pi}{s}} + u \cdot -0.5\right)}\right) \]

   4. associate-*r/ 16.1%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \left(\color{blue}{\frac{u \cdot 1}{2 - \frac{\pi}{s}}} + u \cdot -0.5\right)\right) \]

   5. *-rgt-identity 16.1%

      \[\leadsto \left(-s\right) \cdot \left(-4 \cdot \left(\frac{\color{blue}{u}}{2 - \frac{\pi}{s}} + u \cdot -0.5\right)\right) \]

10. Simplified 16.1%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \left(\frac{u}{2 - \frac{\pi}{s}} + u \cdot -0.5\right)\right)} \]

11. Final simplification 16.1%

    \[\leadsto s \cdot \left(-4 \cdot \left(u \cdot \left(--0.5\right) - \frac{u}{2 - \frac{\pi}{s}}\right)\right) \]

Alternative 10: 16.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-4 \cdot \left(u \cdot \left(0.5 - \frac{1}{2 - \frac{\pi}{s}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (* -4.0 (* u (- 0.5 (/ 1.0 (- 2.0 (/ PI s))))))))

C

float code(float u, float s) {
	return s * (-4.0f * (u * (0.5f - (1.0f / (2.0f - (((float) M_PI) / s))))));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(-4.0) * Float32(u * Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(pi) / s)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (single(-4.0) * (u * (single(0.5) - (single(1.0) / (single(2.0) - (single(pi) / s))))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 8.8%

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

5. Taylor expanded in s around inf 15.4%

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

   1. mul-1-neg 15.4%

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

   2. unsub-neg 15.4%

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

7. Simplified 15.4%

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

8. Taylor expanded in u around 0 16.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \left(u \cdot \left(\frac{1}{2 - \frac{\pi}{s}} - 0.5\right)\right)\right)} \]

9. Final simplification 16.1%

   \[\leadsto s \cdot \left(-4 \cdot \left(u \cdot \left(0.5 - \frac{1}{2 - \frac{\pi}{s}}\right)\right)\right) \]

Alternative 11: 15.5% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -log(((-1.0e0) + (2.0e0 / (1.0e0 - u))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 8.8%

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

5. Taylor expanded in s around inf 15.4%

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

   1. mul-1-neg 15.4%

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

   2. unsub-neg 15.4%

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

7. Simplified 15.4%

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

8. Taylor expanded in s around 0 15.4%

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

   1. sub-neg 15.4%

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

   2. associate-*r/ 15.4%

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

   3. metadata-eval 15.4%

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

   4. metadata-eval 15.4%

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

10. Simplified 15.4%

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

11. Final simplification 15.4%

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

Alternative 12: 11.5% accurate, 6.8× 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 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around -inf 12.5%

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

   1. associate--r+ 12.5%

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

   2. cancel-sign-sub-inv 12.5%

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

   3. associate-*r* 12.5%

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

   4. distribute-rgt-out-- 12.5%

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

   5. *-commutative 12.5%

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

   6. metadata-eval 12.5%

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

   7. *-commutative 12.5%

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

   8. *-commutative 12.5%

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

   9. associate-*l* 12.5%

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

6. Simplified 12.5%

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

7. Taylor expanded in u around 0 12.5%

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

   1. +-commutative 12.5%

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

   2. associate-*r* 12.5%

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

   3. distribute-rgt-out 12.5%

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

   4. *-commutative 12.5%

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

9. Simplified 12.5%

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

10. Final simplification 12.5%

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

Alternative 13: 11.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{-1}{\frac{s}{\pi}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{0 - s \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
6. Step-by-step derivation

   1. neg-sub0 6.8%

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

   2. distribute-rgt-neg-in 6.8%

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

   3. +-commutative 6.8%

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

7. Simplified 6.8%

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

8. Taylor expanded in u around 0 12.3%

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

   1. expm1-log1p-u 12.3%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}\right) \]

10. Applied egg-rr 12.3%

    \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 12.3%

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

    2. clear-num 12.3%

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

12. Applied egg-rr 12.3%

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

13. Final simplification 12.3%

    \[\leadsto s \cdot \frac{-1}{\frac{s}{\pi}} \]

Alternative 14: 11.3% accurate, 7.2× 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 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in u around 0 12.3%

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

   1. neg-mul-1 12.3%

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

6. Simplified 12.3%

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

7. Final simplification 12.3%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023293 
(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))))