Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 18.1s

Alternatives: 9

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: 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

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

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: 11.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{-4}{\frac{s}{{\left(\sqrt[3]{\pi}\right)}^{3} \cdot \left(u \cdot -0.25 + \mathsf{fma}\left(u, -0.25, 0.25\right)\right)}} \end{array} \]

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(Float32(-4.0) / Float32(s / Float32((cbrt(Float32(pi)) ^ Float32(3.0)) * 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 11.5%

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

7. Simplified 11.5%

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

   1. add-log-exp 11.5%

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

9. Applied egg-rr 11.5%

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

    1. distribute-rgt-neg-out 11.5%

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

    2. associate-*r/ 11.5%

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

    3. fma-def 11.5%

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

    4. fma-neg 11.5%

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

    5. metadata-eval 11.5%

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

    6. rem-log-exp 11.5%

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

11. Applied egg-rr 11.5%

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

    1. distribute-rgt-neg-in 11.5%

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

    2. associate-/l* 11.5%

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

    3. distribute-neg-frac 11.5%

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

    4. metadata-eval 11.5%

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

    5. fma-udef 11.5%

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

    6. fma-udef 11.5%

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

    7. *-commutative 11.5%

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

    8. +-commutative 11.5%

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

    9. associate-*r* 11.5%

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

    10. *-commutative 11.5%

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

    11. +-commutative 11.5%

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

    12. associate-*r* 11.5%

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

    13. *-commutative 11.5%

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

    14. distribute-rgt-out 11.5%

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

    15. *-commutative 11.5%

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

    16. +-commutative 11.5%

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

13. Simplified 11.5%

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

    1. add-cube-cbrt 11.5%

       \[\leadsto s \cdot \frac{-4}{\frac{s}{\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(u \cdot -0.25 + \mathsf{fma}\left(u, -0.25, 0.25\right)\right)}} \]

    2. pow3 11.5%

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

15. Applied egg-rr 11.5%

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

16. Final simplification 11.5%

    \[\leadsto s \cdot \frac{-4}{\frac{s}{{\left(\sqrt[3]{\pi}\right)}^{3} \cdot \left(u \cdot -0.25 + \mathsf{fma}\left(u, -0.25, 0.25\right)\right)}} \]
17. Add Preprocessing

Alternative 3: 11.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{s \cdot \left(4 \cdot \frac{\mathsf{fma}\left(u \cdot \pi, 0.5, \pi \cdot -0.25\right)}{s}\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (pow (cbrt (* s (* 4.0 (/ (fma (* u PI) 0.5 (* PI -0.25)) s)))) 3.0))

C

float code(float u, float s) {
	return powf(cbrtf((s * (4.0f * (fmaf((u * ((float) M_PI)), 0.5f, (((float) M_PI) * -0.25f)) / s)))), 3.0f);
}

Julia

function code(u, s)
	return cbrt(Float32(s * Float32(Float32(4.0) * Float32(fma(Float32(u * Float32(pi)), Float32(0.5), Float32(Float32(pi) * Float32(-0.25))) / s)))) ^ Float32(3.0)
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. Taylor expanded in s around inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. distribute-rgt-out-- 11.5%

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

   4. *-commutative 11.5%

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

   5. metadata-eval 11.5%

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

   6. metadata-eval 11.5%

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

   7. *-commutative 11.5%

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

7. Simplified 11.5%

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

   1. add-cube-cbrt 11.5%

      \[\leadsto \color{blue}{\left(\sqrt[3]{s \cdot \left(--4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)} \cdot \sqrt[3]{s \cdot \left(--4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)}\right) \cdot \sqrt[3]{s \cdot \left(--4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)}} \]

   2. pow3 11.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{s \cdot \left(--4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)}\right)}^{3}} \]

   3. distribute-lft-neg-in 11.5%

      \[\leadsto {\left(\sqrt[3]{s \cdot \color{blue}{\left(\left(--4\right) \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)}}\right)}^{3} \]

   4. metadata-eval 11.5%

      \[\leadsto {\left(\sqrt[3]{s \cdot \left(\color{blue}{4} \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right)}\right)}^{3} \]

   5. fma-def 11.5%

      \[\leadsto {\left(\sqrt[3]{s \cdot \left(4 \cdot \frac{\color{blue}{\mathsf{fma}\left(\pi \cdot u, 0.5, \pi \cdot -0.25\right)}}{s}\right)}\right)}^{3} \]

9. Applied egg-rr 11.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{s \cdot \left(4 \cdot \frac{\mathsf{fma}\left(\pi \cdot u, 0.5, \pi \cdot -0.25\right)}{s}\right)}\right)}^{3}} \]

10. Final simplification 11.5%

    \[\leadsto {\left(\sqrt[3]{s \cdot \left(4 \cdot \frac{\mathsf{fma}\left(u \cdot \pi, 0.5, \pi \cdot -0.25\right)}{s}\right)}\right)}^{3} \]
11. Add Preprocessing

Alternative 4: 11.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(\pi \cdot -0.25 + 0.5 \cdot {\left(\sqrt[3]{u \cdot \pi}\right)}^{3}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* 4.0 (+ (* PI -0.25) (* 0.5 (pow (cbrt (* u PI)) 3.0)))))

C

float code(float u, float s) {
	return 4.0f * ((((float) M_PI) * -0.25f) + (0.5f * powf(cbrtf((u * ((float) M_PI))), 3.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(pi) * Float32(-0.25)) + Float32(Float32(0.5) * (cbrt(Float32(u * Float32(pi))) ^ Float32(3.0)))))
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. Taylor expanded in s around inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. distribute-rgt-out-- 11.5%

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

   4. *-commutative 11.5%

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

   5. metadata-eval 11.5%

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

   6. metadata-eval 11.5%

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

   7. *-commutative 11.5%

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

7. Simplified 11.5%

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

   1. add-cube-cbrt 11.5%

      \[\leadsto 4 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi \cdot u} \cdot \sqrt[3]{\pi \cdot u}\right) \cdot \sqrt[3]{\pi \cdot u}\right)} \cdot 0.5 + \pi \cdot -0.25\right) \]

   2. pow3 11.5%

      \[\leadsto 4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi \cdot u}\right)}^{3}} \cdot 0.5 + \pi \cdot -0.25\right) \]

9. Applied egg-rr 11.5%

   \[\leadsto 4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi \cdot u}\right)}^{3}} \cdot 0.5 + \pi \cdot -0.25\right) \]

10. Final simplification 11.5%

    \[\leadsto 4 \cdot \left(\pi \cdot -0.25 + 0.5 \cdot {\left(\sqrt[3]{u \cdot \pi}\right)}^{3}\right) \]
11. Add Preprocessing

Alternative 5: 11.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-4.0) * Float32(s * Float32(Float32(Float32(pi) * Float32(Float32(u * Float32(-0.25)) + fma(u, Float32(-0.25), Float32(0.25)))) / 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 11.5%

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

7. Simplified 11.5%

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

   1. expm1-log1p-u 0.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(-4 \cdot \frac{\pi \cdot \left(u \cdot -0.25 - -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}\right)\right)\right)} \]

   2. expm1-udef 0.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(-4 \cdot \frac{\pi \cdot \left(u \cdot -0.25 - -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}\right)\right)} - 1} \]

   3. distribute-lft-neg-in 0.6%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \color{blue}{\left(\left(-4\right) \cdot \frac{\pi \cdot \left(u \cdot -0.25 - -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}\right)}\right)} - 1 \]

   4. metadata-eval 0.6%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \left(\color{blue}{-4} \cdot \frac{\pi \cdot \left(u \cdot -0.25 - -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}\right)\right)} - 1 \]

   5. fma-def 0.6%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \left(-4 \cdot \frac{\color{blue}{\mathsf{fma}\left(\pi, u \cdot -0.25 - -0.25, u \cdot \left(\pi \cdot -0.25\right)\right)}}{s}\right)\right)} - 1 \]

   6. fma-neg 0.6%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \left(-4 \cdot \frac{\mathsf{fma}\left(\pi, \color{blue}{\mathsf{fma}\left(u, -0.25, --0.25\right)}, u \cdot \left(\pi \cdot -0.25\right)\right)}{s}\right)\right)} - 1 \]

   7. metadata-eval 0.6%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \left(-4 \cdot \frac{\mathsf{fma}\left(\pi, \mathsf{fma}\left(u, -0.25, \color{blue}{0.25}\right), u \cdot \left(\pi \cdot -0.25\right)\right)}{s}\right)\right)} - 1 \]

9. Applied egg-rr 0.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(-4 \cdot \frac{\mathsf{fma}\left(\pi, \mathsf{fma}\left(u, -0.25, 0.25\right), u \cdot \left(\pi \cdot -0.25\right)\right)}{s}\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 0.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(-4 \cdot \frac{\mathsf{fma}\left(\pi, \mathsf{fma}\left(u, -0.25, 0.25\right), u \cdot \left(\pi \cdot -0.25\right)\right)}{s}\right)\right)\right)} \]

    2. expm1-log1p 11.5%

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

    3. *-commutative 11.5%

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

    4. associate-*l* 11.5%

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

11. Simplified 11.5%

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

12. Final simplification 11.5%

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

Alternative 6: 11.5% 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. Step-by-step derivation

   1. div-inv 98.9%

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

   2. add-cube-cbrt 98.9%

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

   3. associate-*l* 98.9%

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

   4. pow2 98.9%

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in s around -inf 11.5%

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

9. Simplified 11.5%

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

10. Final simplification 11.5%

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

Alternative 7: 11.5% 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%

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

   1. div-inv 98.9%

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

   2. add-cube-cbrt 98.9%

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

   3. associate-*l* 98.9%

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

   4. pow2 98.9%

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in s around inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. *-commutative 11.5%

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

   4. *-commutative 11.5%

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

   5. distribute-rgt-out-- 11.5%

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

   6. metadata-eval 11.5%

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

   7. metadata-eval 11.5%

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

   8. *-commutative 11.5%

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

   9. distribute-rgt-in 11.5%

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

   10. associate-*l* 11.5%

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

   11. metadata-eval 11.5%

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

   12. *-commutative 11.5%

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

   13. associate-*l* 11.5%

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

   14. metadata-eval 11.5%

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

   15. *-commutative 11.5%

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

9. Simplified 11.5%

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

10. Final simplification 11.5%

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

Alternative 8: 11.5% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (u * single(pi))) - 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 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. distribute-rgt-out-- 11.5%

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

   4. *-commutative 11.5%

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

   5. metadata-eval 11.5%

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

   6. metadata-eval 11.5%

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

   7. *-commutative 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in u around 0 11.5%

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

9. Final simplification 11.5%

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

Alternative 9: 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%

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

5. Taylor expanded in u around 0 11.3%

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

   1. neg-mul-1 11.3%

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

7. Simplified 11.3%

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

8. Final simplification 11.3%

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

Reproduce

herbie shell --seed 2024018 
(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))))