Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 11.6s

Alternatives: 11

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 13.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.0%

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

   1. *-un-lft-identity 11.0%

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

   2. add-sqr-sqrt 11.0%

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

   3. times-frac 11.0%

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

7. Applied egg-rr 11.0%

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

   1. associate-*l/ 11.0%

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

   2. *-lft-identity 11.0%

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

9. Simplified 11.0%

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

    1. add-log-exp 11.0%

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

    2. exp-prod 13.2%

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

    3. add-sqr-sqrt -0.0%

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

    4. sqrt-unprod 10.2%

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

    5. sqr-neg 10.2%

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

    6. sqrt-unprod 10.2%

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

    7. add-sqr-sqrt 10.2%

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

    8. associate-/l/ 10.2%

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

    9. add-sqr-sqrt 10.2%

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

11. Applied egg-rr 10.2%

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

    1. add-sqr-sqrt 10.2%

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

    2. sqrt-unprod 10.2%

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

    3. sqr-neg 10.2%

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

    4. sqrt-unprod 9.5%

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

    5. add-sqr-sqrt 13.2%

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

    6. frac-2neg 13.2%

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

    7. div-inv 13.2%

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

    8. remove-double-neg 13.2%

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

13. Applied egg-rr 13.2%

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

14. Final simplification 13.2%

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

Alternative 3: 13.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.0%

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

   1. *-un-lft-identity 11.0%

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

   2. add-sqr-sqrt 11.0%

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

   3. times-frac 11.0%

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

7. Applied egg-rr 11.0%

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

   1. associate-*l/ 11.0%

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

   2. *-lft-identity 11.0%

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

9. Simplified 11.0%

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

    1. add-log-exp 11.0%

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

    2. exp-prod 13.2%

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

    3. add-sqr-sqrt -0.0%

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

    4. sqrt-unprod 10.2%

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

    5. sqr-neg 10.2%

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

    6. sqrt-unprod 10.2%

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

    7. add-sqr-sqrt 10.2%

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

    8. associate-/l/ 10.2%

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

    9. add-sqr-sqrt 10.2%

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

11. Applied egg-rr 10.2%

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

    1. add-sqr-sqrt 10.2%

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

    2. sqrt-unprod 10.2%

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

    3. sqr-neg 10.2%

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

    4. sqrt-unprod 9.5%

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

    5. add-sqr-sqrt 13.2%

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

    6. distribute-frac-neg2 13.2%

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

13. Applied egg-rr 13.2%

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

14. Final simplification 13.2%

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

Alternative 4: 11.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\pi \cdot \left({u}^{3} \cdot -0.015625 + 0.015625\right)}{0.0625}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  4.0
  (-
   (* 0.25 (* u PI))
   (/ (* PI (+ (* (pow u 3.0) -0.015625) 0.015625)) 0.0625))))

C

float code(float u, float s) {
	return 4.0f * ((0.25f * (u * ((float) M_PI))) - ((((float) M_PI) * ((powf(u, 3.0f) * -0.015625f) + 0.015625f)) / 0.0625f));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) - Float32(Float32(Float32(pi) * Float32(Float32((u ^ Float32(3.0)) * Float32(-0.015625)) + Float32(0.015625))) / Float32(0.0625))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.25) * (u * single(pi))) - ((single(pi) * (((u ^ single(3.0)) * single(-0.015625)) + single(0.015625))) / single(0.0625)));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 11.2%

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

   1. associate-*r* 11.2%

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

   2. *-commutative 11.2%

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

   3. distribute-rgt-in 11.2%

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

   4. metadata-eval 11.2%

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

   5. sub-neg 11.2%

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

   6. *-commutative 11.2%

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

   7. flip3-- 11.2%

      \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \color{blue}{\frac{{\left(u \cdot -0.25\right)}^{3} - {-0.25}^{3}}{\left(u \cdot -0.25\right) \cdot \left(u \cdot -0.25\right) + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}} \cdot \pi\right) \]

   8. associate-*l/ 11.2%

      \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \color{blue}{\frac{\left({\left(u \cdot -0.25\right)}^{3} - {-0.25}^{3}\right) \cdot \pi}{\left(u \cdot -0.25\right) \cdot \left(u \cdot -0.25\right) + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}}\right) \]

   9. sub-neg 11.2%

      \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\color{blue}{\left({\left(u \cdot -0.25\right)}^{3} + \left(-{-0.25}^{3}\right)\right)} \cdot \pi}{\left(u \cdot -0.25\right) \cdot \left(u \cdot -0.25\right) + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   10. unpow-prod-down 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left(\color{blue}{{u}^{3} \cdot {-0.25}^{3}} + \left(-{-0.25}^{3}\right)\right) \cdot \pi}{\left(u \cdot -0.25\right) \cdot \left(u \cdot -0.25\right) + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   11. metadata-eval 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot \color{blue}{-0.015625} + \left(-{-0.25}^{3}\right)\right) \cdot \pi}{\left(u \cdot -0.25\right) \cdot \left(u \cdot -0.25\right) + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   12. metadata-eval 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot -0.015625 + \left(-\color{blue}{-0.015625}\right)\right) \cdot \pi}{\left(u \cdot -0.25\right) \cdot \left(u \cdot -0.25\right) + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   13. metadata-eval 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot -0.015625 + \color{blue}{0.015625}\right) \cdot \pi}{\left(u \cdot -0.25\right) \cdot \left(u \cdot -0.25\right) + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   14. swap-sqr 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot -0.015625 + 0.015625\right) \cdot \pi}{\color{blue}{\left(u \cdot u\right) \cdot \left(-0.25 \cdot -0.25\right)} + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   15. pow2 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot -0.015625 + 0.015625\right) \cdot \pi}{\color{blue}{{u}^{2}} \cdot \left(-0.25 \cdot -0.25\right) + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   16. metadata-eval 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot -0.015625 + 0.015625\right) \cdot \pi}{{u}^{2} \cdot \color{blue}{0.0625} + \left(-0.25 \cdot -0.25 + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   17. metadata-eval 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot -0.015625 + 0.015625\right) \cdot \pi}{{u}^{2} \cdot 0.0625 + \left(\color{blue}{0.0625} + \left(u \cdot -0.25\right) \cdot -0.25\right)}\right) \]

   18. associate-*l* 11.2%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot -0.015625 + 0.015625\right) \cdot \pi}{{u}^{2} \cdot 0.0625 + \left(0.0625 + \color{blue}{u \cdot \left(-0.25 \cdot -0.25\right)}\right)}\right) \]

7. Applied egg-rr 11.2%

   \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \color{blue}{\frac{\left({u}^{3} \cdot -0.015625 + 0.015625\right) \cdot \pi}{{u}^{2} \cdot 0.0625 + \left(0.0625 + u \cdot 0.0625\right)}}\right) \]

8. Taylor expanded in u around 0 11.2%

   \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\left({u}^{3} \cdot -0.015625 + 0.015625\right) \cdot \pi}{\color{blue}{0.0625}}\right) \]

9. Final simplification 11.2%

   \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \frac{\pi \cdot \left({u}^{3} \cdot -0.015625 + 0.015625\right)}{0.0625}\right) \]
10. Add Preprocessing

Alternative 5: 11.6% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 11.2%

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

6. Final simplification 11.2%

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

Alternative 6: 11.6% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 11.2%

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

6. Taylor expanded in u around inf 11.2%

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

7. Final simplification 11.2%

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

Alternative 7: 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. associate-*r* 11.2%

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

   4. distribute-rgt-out-- 11.2%

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

   5. *-commutative 11.2%

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

   6. metadata-eval 11.2%

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

   7. *-commutative 11.2%

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

   8. *-commutative 11.2%

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

   9. associate-*l* 11.2%

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

7. Simplified 11.2%

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

8. Taylor expanded in u around inf 11.2%

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

   1. +-commutative 11.2%

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

   2. mul-1-neg 11.2%

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

   3. unsub-neg 11.2%

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

   4. *-commutative 11.2%

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

10. Simplified 11.2%

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

Alternative 8: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. associate-*r* 11.2%

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

   4. distribute-rgt-out-- 11.2%

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

   5. *-commutative 11.2%

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

   6. metadata-eval 11.2%

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

   7. *-commutative 11.2%

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

   8. *-commutative 11.2%

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

   9. associate-*l* 11.2%

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

7. Simplified 11.2%

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

8. Taylor expanded in u around 0 11.2%

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

   1. neg-mul-1 11.2%

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

   2. +-commutative 11.2%

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

   3. associate-*r* 11.2%

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

   4. neg-mul-1 11.2%

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

   5. distribute-rgt-out 11.2%

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

   6. *-commutative 11.2%

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

10. Simplified 11.2%

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

11. Final simplification 11.2%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]
12. Add Preprocessing

Alternative 9: 11.3% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.0%

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

   1. *-un-lft-identity 11.0%

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

   2. add-sqr-sqrt 11.0%

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

   3. times-frac 11.0%

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

7. Applied egg-rr 11.0%

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

   1. associate-*l/ 11.0%

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

   2. *-lft-identity 11.0%

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

9. Simplified 11.0%

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

    1. associate-*r/ 11.0%

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

    2. add-sqr-sqrt -0.0%

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

    3. sqrt-unprod 8.7%

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

    4. sqr-neg 8.7%

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

    5. sqrt-unprod 4.7%

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

    6. add-sqr-sqrt 4.7%

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

11. Applied egg-rr 4.7%

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

    1. associate-*r/ 4.7%

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

    2. associate-/l/ 4.7%

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

    3. rem-square-sqrt 4.7%

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

    4. associate-*r/ 4.7%

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

13. Simplified 4.7%

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

    1. add-sqr-sqrt 4.7%

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

    2. sqrt-unprod 8.5%

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

    3. sqr-neg 8.5%

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

    4. sqrt-unprod -0.0%

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

    5. add-sqr-sqrt 11.0%

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

    6. distribute-lft-neg-out 11.0%

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

    7. clear-num 11.0%

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

    8. un-div-inv 11.0%

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

15. Applied egg-rr 11.0%

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

16. Final simplification 11.0%

    \[\leadsto \frac{s}{\frac{-s}{\pi}} \]
17. Add Preprocessing

Alternative 10: 11.3% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.0%

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

   1. neg-mul-1 11.0%

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

7. Simplified 11.0%

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

Alternative 11: 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%

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

5. Taylor expanded in u around 0 11.0%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 8.5%

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

   3. sqr-neg 8.5%

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

   4. sqrt-unprod 4.7%

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

   5. add-sqr-sqrt 4.7%

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

   6. clear-num 4.7%

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

   7. un-div-inv 4.7%

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

7. Applied egg-rr 4.7%

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

   1. associate-/r/ 4.7%

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

   2. *-inverses 4.7%

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

   3. *-lft-identity 4.7%

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

9. Simplified 4.7%

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

Reproduce

herbie shell --seed 2024108 
(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))))