Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.3s

Alternatives: 10

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 37.8%

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

   1. *-un-lft-identity 37.8%

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

   2. exp-prod 37.8%

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

7. Applied egg-rr 37.8%

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

   1. exp-1-e 37.8%

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

9. Simplified 37.8%

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

10. Final simplification 37.8%

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

Alternative 3: 37.6% accurate, 2.0× 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 37.8%

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

   1. *-un-lft-identity 37.8%

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

   2. exp-prod 37.8%

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

7. Applied egg-rr 37.8%

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

   1. exp-1-e 37.8%

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

9. Simplified 37.8%

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

    1. e-exp-1 37.8%

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

    2. add-sqr-sqrt 37.8%

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

    3. sqrt-unprod 37.8%

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

    4. sqr-neg 37.8%

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

    5. sqrt-unprod -0.0%

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

    6. add-sqr-sqrt 6.4%

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

    7. exp-prod 6.4%

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

    8. *-un-lft-identity 6.4%

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

    9. exp-neg 6.4%

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

11. Applied egg-rr 6.4%

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

    1. rec-exp 6.4%

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

    2. distribute-neg-frac 6.4%

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

13. Simplified 6.4%

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

    1. +-commutative 6.4%

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

    2. *-un-lft-identity 6.4%

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

    3. fma-def 6.4%

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

    4. clear-num 6.4%

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

    5. clear-num 6.4%

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

    6. add-sqr-sqrt -0.0%

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

    7. sqrt-unprod 37.8%

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

    8. sqr-neg 37.8%

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

    9. sqrt-unprod 37.8%

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

    10. add-sqr-sqrt 37.8%

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

    11. div-inv 37.8%

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

    12. metadata-eval 37.8%

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

    13. metadata-eval 37.8%

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

15. Applied egg-rr 37.8%

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

    1. fma-udef 37.8%

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

    2. *-lft-identity 37.8%

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

    3. *-commutative 37.8%

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

17. Simplified 37.8%

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

18. Final simplification 37.8%

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

Alternative 4: 36.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 37.8%

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

6. Taylor expanded in s around inf 35.9%

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

7. Final simplification 35.9%

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

Alternative 5: 11.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 12.0%

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

   1. sub-neg 12.0%

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

   2. associate-*r* 12.0%

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

   3. *-commutative 12.0%

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

   4. distribute-rgt-in 12.0%

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

   5. fma-def 12.0%

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

7. Applied egg-rr 12.0%

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

   1. sub-neg 12.0%

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

   2. *-commutative 12.0%

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

   3. associate-*r* 12.0%

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

   4. distribute-rgt-out-- 12.0%

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

   5. *-commutative 12.0%

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

9. Simplified 12.0%

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

10. Final simplification 12.0%

    \[\leadsto 4 \cdot \left(\pi \cdot \left(u \cdot 0.25 - \mathsf{fma}\left(u, -0.25, 0.25\right)\right)\right) \]
11. Add Preprocessing

Alternative 6: 11.7% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 12.0%

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

   1. associate--r+ 12.0%

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

   2. cancel-sign-sub-inv 12.0%

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

   3. metadata-eval 12.0%

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

   4. cancel-sign-sub-inv 12.0%

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

   5. associate-*r* 12.0%

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

   6. distribute-rgt-out 12.0%

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

   7. *-commutative 12.0%

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

   8. metadata-eval 12.0%

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

   9. *-commutative 12.0%

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

   10. associate-*l* 12.0%

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

7. Simplified 12.0%

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

8. Final simplification 12.0%

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

Alternative 7: 11.7% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 12.0%

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

   1. associate--r+ 12.0%

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

   2. cancel-sign-sub-inv 12.0%

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

   3. distribute-rgt-out-- 12.0%

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

   4. *-commutative 12.0%

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

   5. metadata-eval 12.0%

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

   6. metadata-eval 12.0%

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

   7. *-commutative 12.0%

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

7. Simplified 12.0%

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

8. Final simplification 12.0%

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

Alternative 8: 11.7% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 12.0%

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

   1. associate--r+ 12.0%

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

   2. cancel-sign-sub-inv 12.0%

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

   3. distribute-rgt-out-- 12.0%

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

   4. *-commutative 12.0%

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

   5. metadata-eval 12.0%

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

   6. metadata-eval 12.0%

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

   7. *-commutative 12.0%

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

7. Simplified 12.0%

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

8. Taylor expanded in u around 0 12.0%

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

   1. +-commutative 12.0%

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

   2. associate-*r* 12.0%

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

   3. distribute-rgt-out 12.0%

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

10. Simplified 12.0%

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

11. Final simplification 12.0%

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

Alternative 9: 11.4% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.8%

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

   1. neg-mul-1 11.8%

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

7. Simplified 11.8%

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

8. Final simplification 11.8%

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

Alternative 10: 4.6% accurate, 433.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 PI)

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 37.8%

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

6. Taylor expanded in u around 0 11.8%

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

   1. expm1-log1p-u -0.0%

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

   2. expm1-udef -0.0%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 3.2%

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

   5. sqr-neg 3.2%

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

   6. sqrt-unprod 4.5%

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

   7. add-sqr-sqrt 4.5%

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

8. Applied egg-rr 4.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot \frac{\pi}{s}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 4.5%

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

   2. expm1-log1p 4.5%

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

10. Simplified 4.5%

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

11. Taylor expanded in s around 0 4.5%

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

12. Final simplification 4.5%

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

Reproduce

herbie shell --seed 2024029 
(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))))