Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 28.3s

Alternatives: 15

Speedup: 0.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\\ s \cdot \log \left(\frac{1 + \left({t\_0}^{-2} + \frac{1}{t\_0}\right)}{-1 + {t\_0}^{-3}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0
         (+
          (/ u (+ 1.0 (exp (/ PI (- s)))))
          (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))))
   (*
    s
    (log (/ (+ 1.0 (+ (pow t_0 -2.0) (/ 1.0 t_0))) (+ -1.0 (pow t_0 -3.0)))))))

C

float code(float u, float s) {
	float t_0 = (u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s))));
	return s * logf(((1.0f + (powf(t_0, -2.0f) + (1.0f / t_0))) / (-1.0f + powf(t_0, -3.0f))));
}

Julia

function code(u, s)
	t_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)))))
	return Float32(s * log(Float32(Float32(Float32(1.0) + Float32((t_0 ^ Float32(-2.0)) + Float32(Float32(1.0) / t_0))) / Float32(Float32(-1.0) + (t_0 ^ Float32(-3.0))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = (u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s))));
	tmp = s * log(((single(1.0) + ((t_0 ^ single(-2.0)) + (single(1.0) / t_0))) / (single(-1.0) + (t_0 ^ single(-3.0)))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. add-cube-cbrt 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\left(\sqrt[3]{\frac{\pi}{s}} \cdot \sqrt[3]{\frac{\pi}{s}}\right) \cdot \sqrt[3]{\frac{\pi}{s}}}}}} + -1\right) \]

   2. pow3 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right) \]

6. Applied egg-rr 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right) \]
7. Step-by-step derivation

   1. flip3-+ 98.7%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}\right)}^{3} + {-1}^{3}}{\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} \cdot \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} + \left(-1 \cdot -1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} \cdot -1\right)}\right)} \]

8. Applied egg-rr 98.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3} + -1}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1\right)}\right)} \]
9. Step-by-step derivation

   1. +-commutative 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{\color{blue}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1\right)}\right) \]

   2. associate-*l/ 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \color{blue}{\frac{1 \cdot -1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right)}\right) \]

   3. metadata-eval 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{\color{blue}{-1}}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}\right) \]

10. Simplified 98.9%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}\right)} \]
11. Step-by-step derivation

    1. clear-num 98.8%

       \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{1}{\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}}\right)} \]

    2. log-rec 98.9%

       \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-\log \left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)\right)} \]

12. Applied egg-rr 98.9%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-\log \left(\frac{\left({\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + 1\right) - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)\right)} \]
13. Step-by-step derivation

    1. pow1 98.9%

       \[\leadsto \color{blue}{{\left(\left(-s\right) \cdot \left(-\log \left(\frac{\left({\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + 1\right) - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)\right)\right)}^{1}} \]

14. Applied egg-rr 98.9%

    \[\leadsto \color{blue}{{\left(\left(-s\right) \cdot \left(-\log \left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)\right)\right)}^{1}} \]
15. Step-by-step derivation

    1. unpow1 98.9%

       \[\leadsto \color{blue}{\left(-s\right) \cdot \left(-\log \left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)\right)} \]

    2. distribute-rgt-neg-in 98.9%

       \[\leadsto \color{blue}{-\left(-s\right) \cdot \log \left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)} \]

    3. distribute-lft-neg-in 98.9%

       \[\leadsto \color{blue}{\left(-\left(-s\right)\right) \cdot \log \left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)} \]

16. Simplified 98.9%

    \[\leadsto \color{blue}{\left(-\left(-s\right)\right) \cdot \log \left(\frac{1 + \left({\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)} \]

17. Final simplification 98.9%

    \[\leadsto s \cdot \log \left(\frac{1 + \left({\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right) \]
18. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{-1 + {t\_0}^{-3}}{{t\_0}^{-2} + \left(1 + \frac{1}{t\_0}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0
         (+
          (/ u (+ 1.0 (exp (/ PI (- s)))))
          (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))))
   (*
    (- s)
    (log (/ (+ -1.0 (pow t_0 -3.0)) (+ (pow t_0 -2.0) (+ 1.0 (/ 1.0 t_0))))))))

C

float code(float u, float s) {
	float t_0 = (u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s))));
	return -s * logf(((-1.0f + powf(t_0, -3.0f)) / (powf(t_0, -2.0f) + (1.0f + (1.0f / t_0)))));
}

Julia

function code(u, s)
	t_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)))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(-1.0) + (t_0 ^ Float32(-3.0))) / Float32((t_0 ^ Float32(-2.0)) + Float32(Float32(1.0) + Float32(Float32(1.0) / t_0))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = (u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s))));
	tmp = -s * log(((single(-1.0) + (t_0 ^ single(-3.0))) / ((t_0 ^ single(-2.0)) + (single(1.0) + (single(1.0) / t_0)))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. add-cube-cbrt 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\left(\sqrt[3]{\frac{\pi}{s}} \cdot \sqrt[3]{\frac{\pi}{s}}\right) \cdot \sqrt[3]{\frac{\pi}{s}}}}}} + -1\right) \]

   2. pow3 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right) \]

6. Applied egg-rr 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right) \]
7. Step-by-step derivation

   1. flip3-+ 98.7%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}\right)}^{3} + {-1}^{3}}{\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} \cdot \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} + \left(-1 \cdot -1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} \cdot -1\right)}\right)} \]

8. Applied egg-rr 98.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3} + -1}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1\right)}\right)} \]
9. Step-by-step derivation

   1. +-commutative 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{\color{blue}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1\right)}\right) \]

   2. associate-*l/ 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \color{blue}{\frac{1 \cdot -1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right)}\right) \]

   3. metadata-eval 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{\color{blue}{-1}}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}\right) \]

10. Simplified 98.9%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}\right)} \]

11. Final simplification 98.9%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}\right) \]
12. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s))))
   (*
    (- s)
    (log
     (+
      -1.0
      (/
       1.0
       (+
        (/ 1.0 (+ 1.0 t_0))
        (+ (/ u (+ 1.0 (exp (/ PI (- s))))) (/ u (- -1.0 t_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 / (1.0f + t_0)) + ((u / (1.0f + expf((((float) M_PI) / -s)))) + (u / (-1.0f - t_0)))))));
}

Julia

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around 0 98.8%

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

   1. associate-*r* 98.8%

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

   2. mul-1-neg 98.8%

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

   3. sub-neg 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log1p
   (+
    -1.0
    (+
     -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 * log1pf((-1.0f + (-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) * log1p(Float32(Float32(-1.0) + 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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. add-cube-cbrt 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\left(\sqrt[3]{\frac{\pi}{s}} \cdot \sqrt[3]{\frac{\pi}{s}}\right) \cdot \sqrt[3]{\frac{\pi}{s}}}}}} + -1\right) \]

   2. pow3 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right) \]

6. Applied egg-rr 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right) \]
7. Step-by-step derivation

   1. log1p-expm1-u 98.8%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} + -1\right)\right)\right)} \]

   2. expm1-undefine 98.8%

      \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} + -1\right)} - 1}\right) \]

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 5: 99.0% accurate, 1.3× 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 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 6: 25.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 24.3%

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

   1. +-commutative 24.3%

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

   2. fma-define 24.3%

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

7. Simplified 24.3%

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

8. Final simplification 24.3%

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

Alternative 7: 25.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.3%

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

6. Final simplification 24.3%

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

Alternative 8: 14.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{{s}^{2}}{s} \cdot \left(-4 \cdot \frac{\pi \cdot \left(--0.25\right) - \left(u \cdot \pi\right) \cdot 0.5}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (/ (pow s 2.0) s) (* -4.0 (/ (- (* PI (- -0.25)) (* (* u PI) 0.5)) s))))

C

float code(float u, float s) {
	return (powf(s, 2.0f) / s) * (-4.0f * (((((float) M_PI) * -(-0.25f)) - ((u * ((float) M_PI)) * 0.5f)) / s));
}

Julia

function code(u, s)
	return Float32(Float32((s ^ Float32(2.0)) / s) * Float32(Float32(-4.0) * Float32(Float32(Float32(Float32(pi) * Float32(-Float32(-0.25))) - Float32(Float32(u * Float32(pi)) * Float32(0.5))) / s)))
end

MATLAB

function tmp = code(u, s)
	tmp = ((s ^ single(2.0)) / s) * (single(-4.0) * (((single(pi) * -single(-0.25)) - ((u * single(pi)) * single(0.5))) / s));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. distribute-rgt-out-- 11.8%

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

   4. *-commutative 11.8%

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

   5. metadata-eval 11.8%

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

   6. metadata-eval 11.8%

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

   7. *-commutative 11.8%

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

7. Simplified 11.8%

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

   1. neg-sub0 11.8%

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

   2. flip-- 14.1%

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

   3. metadata-eval 14.1%

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

   4. pow2 14.1%

      \[\leadsto \frac{0 - \color{blue}{{s}^{2}}}{0 + s} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   5. add-sqr-sqrt 14.1%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   6. sqrt-unprod 9.2%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   7. sqr-neg 9.2%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   8. sqrt-unprod -0.0%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   9. add-sqr-sqrt 7.3%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   10. sub-neg 7.3%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{0 - s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   11. neg-sub0 7.3%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{-s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   12. add-sqr-sqrt -0.0%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   13. sqrt-unprod 9.2%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   14. sqr-neg 9.2%

       \[\leadsto \frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   15. sqrt-unprod 14.1%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   16. add-sqr-sqrt 14.1%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

9. Applied egg-rr 14.1%

   \[\leadsto \color{blue}{\frac{0 - {s}^{2}}{s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]
10. Step-by-step derivation

    1. sub0-neg 14.1%

       \[\leadsto \frac{\color{blue}{-{s}^{2}}}{s} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

11. Simplified 14.1%

    \[\leadsto \color{blue}{\frac{-{s}^{2}}{s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

12. Final simplification 14.1%

    \[\leadsto \frac{{s}^{2}}{s} \cdot \left(-4 \cdot \frac{\pi \cdot \left(--0.25\right) - \left(u \cdot \pi\right) \cdot 0.5}{s}\right) \]
13. Add Preprocessing

Alternative 9: 11.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(u \cdot \pi\right) \cdot -0.25\\ t_1 := \pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.25\\ t_2 := t\_0 - t\_1\\ t_3 := t\_1 - t\_0\\ -9.142857142857142 \cdot \left(0.25 \cdot t\_2 + 0.5 \cdot t\_2\right) + -0.14285714285714285 \cdot \left(4 \cdot t\_3 + 16 \cdot t\_3\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* (* u PI) -0.25))
        (t_1 (+ (* PI -0.25) (* (* u PI) 0.25)))
        (t_2 (- t_0 t_1))
        (t_3 (- t_1 t_0)))
   (+
    (* -9.142857142857142 (+ (* 0.25 t_2) (* 0.5 t_2)))
    (* -0.14285714285714285 (+ (* 4.0 t_3) (* 16.0 t_3))))))

C

float code(float u, float s) {
	float t_0 = (u * ((float) M_PI)) * -0.25f;
	float t_1 = (((float) M_PI) * -0.25f) + ((u * ((float) M_PI)) * 0.25f);
	float t_2 = t_0 - t_1;
	float t_3 = t_1 - t_0;
	return (-9.142857142857142f * ((0.25f * t_2) + (0.5f * t_2))) + (-0.14285714285714285f * ((4.0f * t_3) + (16.0f * t_3)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(u * Float32(pi)) * Float32(-0.25))
	t_1 = Float32(Float32(Float32(pi) * Float32(-0.25)) + Float32(Float32(u * Float32(pi)) * Float32(0.25)))
	t_2 = Float32(t_0 - t_1)
	t_3 = Float32(t_1 - t_0)
	return Float32(Float32(Float32(-9.142857142857142) * Float32(Float32(Float32(0.25) * t_2) + Float32(Float32(0.5) * t_2))) + Float32(Float32(-0.14285714285714285) * Float32(Float32(Float32(4.0) * t_3) + Float32(Float32(16.0) * t_3))))
end

MATLAB

function tmp = code(u, s)
	t_0 = (u * single(pi)) * single(-0.25);
	t_1 = (single(pi) * single(-0.25)) + ((u * single(pi)) * single(0.25));
	t_2 = t_0 - t_1;
	t_3 = t_1 - t_0;
	tmp = (single(-9.142857142857142) * ((single(0.25) * t_2) + (single(0.5) * t_2))) + (single(-0.14285714285714285) * ((single(4.0) * t_3) + (single(16.0) * t_3)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. add-cube-cbrt 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\left(\sqrt[3]{\frac{\pi}{s}} \cdot \sqrt[3]{\frac{\pi}{s}}\right) \cdot \sqrt[3]{\frac{\pi}{s}}}}}} + -1\right) \]

   2. pow3 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right) \]

6. Applied egg-rr 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right) \]
7. Step-by-step derivation

   1. flip3-+ 98.7%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}\right)}^{3} + {-1}^{3}}{\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} \cdot \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} + \left(-1 \cdot -1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}} \cdot -1\right)}\right)} \]

8. Applied egg-rr 98.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3} + -1}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1\right)}\right)} \]
9. Step-by-step derivation

   1. +-commutative 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{\color{blue}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1\right)}\right) \]

   2. associate-*l/ 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \color{blue}{\frac{1 \cdot -1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right)}\right) \]

   3. metadata-eval 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{\color{blue}{-1}}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}\right) \]

10. Simplified 98.9%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}\right)} \]

11. Taylor expanded in s around -inf 11.8%

    \[\leadsto \color{blue}{-9.142857142857142 \cdot \left(0.25 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right) + 0.5 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)\right) - -0.14285714285714285 \cdot \left(4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right) + 16 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)\right)} \]

12. Final simplification 11.8%

    \[\leadsto -9.142857142857142 \cdot \left(0.25 \cdot \left(\left(u \cdot \pi\right) \cdot -0.25 - \left(\pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.25\right)\right) + 0.5 \cdot \left(\left(u \cdot \pi\right) \cdot -0.25 - \left(\pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.25\right)\right)\right) + -0.14285714285714285 \cdot \left(4 \cdot \left(\left(\pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.25\right) - \left(u \cdot \pi\right) \cdot -0.25\right) + 16 \cdot \left(\left(\pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.25\right) - \left(u \cdot \pi\right) \cdot -0.25\right)\right) \]
13. Add Preprocessing

Alternative 10: 11.6% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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. Final simplification 11.8%

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

Alternative 11: 11.6% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

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

   3. cancel-sign-sub-inv 11.8%

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

   4. metadata-eval 11.8%

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

   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. *-commutative 11.8%

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

   9. *-commutative 11.8%

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

   10. associate-*l* 11.8%

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

7. Simplified 11.8%

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

8. Final simplification 11.8%

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

Alternative 12: 11.6% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. distribute-rgt-out-- 11.8%

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

   4. *-commutative 11.8%

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

   5. metadata-eval 11.8%

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

   6. metadata-eval 11.8%

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

   7. *-commutative 11.8%

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

7. Simplified 11.8%

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

8. Taylor expanded in u around inf 11.8%

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

   1. +-commutative 11.8%

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

   2. mul-1-neg 11.8%

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

   3. unsub-neg 11.8%

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

   4. *-commutative 11.8%

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

10. Simplified 11.8%

    \[\leadsto \color{blue}{u \cdot \left(\pi \cdot 2 - \frac{\pi}{u}\right)} \]
11. Add Preprocessing

Alternative 13: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. distribute-rgt-out-- 11.8%

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

   4. *-commutative 11.8%

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

   5. metadata-eval 11.8%

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

   6. metadata-eval 11.8%

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

   7. *-commutative 11.8%

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

7. Simplified 11.8%

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

8. Taylor expanded in u around 0 11.8%

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

   1. neg-mul-1 11.8%

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

   2. +-commutative 11.8%

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

   3. unsub-neg 11.8%

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

   4. *-commutative 11.8%

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

10. Simplified 11.8%

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

11. Final simplification 11.8%

    \[\leadsto \left(u \cdot \pi\right) \cdot 2 - \pi \]
12. Add Preprocessing

Alternative 14: 11.4% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 11.4%

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

   1. neg-mul-1 11.4%

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

7. Simplified 11.4%

   \[\leadsto \color{blue}{-\pi} \]
8. Add Preprocessing

Alternative 15: 10.4% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 10.2%

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

6. Taylor expanded in s around 0 10.2%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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))))