Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 18.2s

Alternatives: 13

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, 0.9× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (exp (/ (/ PI (cbrt s)) (pow (cbrt s) 2.0)))))))
    -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) / cbrtf(s)) / powf(cbrtf(s), 2.0f))))))) + -1.0f));
}

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

   1. add-cube-cbrt 99.0%

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

   2. *-un-lft-identity 99.0%

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

   3. times-frac 99.0%

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

   4. pow2 99.0%

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

5. Applied egg-rr 99.0%

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

   1. associate-*l/ 99.0%

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

   2. *-lft-identity 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 3: 25.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. +-commutative 24.8%

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

   2. mul-1-neg 24.8%

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

   3. unsub-neg 24.8%

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

   4. associate-*r/ 24.8%

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

   5. *-commutative 24.8%

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

   6. +-commutative 24.8%

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

   7. log1p-def 24.8%

      \[\leadsto \frac{2 \cdot \left(\pi \cdot u\right)}{\frac{\pi}{s} + 1} - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Simplified 24.8%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\pi \cdot u\right)}{\frac{\pi}{s} + 1} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Taylor expanded in s around 0 24.9%

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

    1. mul-1-neg 24.9%

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

12. Simplified 24.9%

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

13. Final simplification 24.9%

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

Alternative 4: 25.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. +-commutative 24.8%

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

   2. mul-1-neg 24.8%

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

   3. unsub-neg 24.8%

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

   4. associate-*r/ 24.8%

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

   5. *-commutative 24.8%

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

   6. +-commutative 24.8%

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

   7. log1p-def 24.8%

      \[\leadsto \frac{2 \cdot \left(\pi \cdot u\right)}{\frac{\pi}{s} + 1} - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Simplified 24.8%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\pi \cdot u\right)}{\frac{\pi}{s} + 1} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Taylor expanded in s around 0 24.9%

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

    1. mul-1-neg 24.9%

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

12. Simplified 24.9%

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

13. Taylor expanded in s around 0 24.9%

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

    1. *-commutative 24.8%

       \[\leadsto 2 \cdot \color{blue}{\left(u \cdot s\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

15. Simplified 24.9%

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

16. Final simplification 24.9%

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

Alternative 5: 25.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

   2. associate-*r* 24.8%

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

   3. neg-mul-1 24.8%

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

9. Simplified 24.8%

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

10. Taylor expanded in s around 0 24.9%

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

    1. mul-1-neg 24.9%

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

12. Simplified 24.9%

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

13. Final simplification 24.9%

    \[\leadsto s \cdot \left(\log s - \log \pi\right) \]

Alternative 6: 25.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(s \cdot u\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- (* 2.0 (* s u)) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return (2.0f * (s * u)) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(2.0) * Float32(s * u)) - Float32(s * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. +-commutative 24.8%

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

   2. mul-1-neg 24.8%

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

   3. unsub-neg 24.8%

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

   4. associate-*r/ 24.8%

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

   5. *-commutative 24.8%

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

   6. +-commutative 24.8%

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

   7. log1p-def 24.8%

      \[\leadsto \frac{2 \cdot \left(\pi \cdot u\right)}{\frac{\pi}{s} + 1} - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Simplified 24.8%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\pi \cdot u\right)}{\frac{\pi}{s} + 1} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Taylor expanded in s around 0 24.8%

    \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
11. Step-by-step derivation

    1. *-commutative 24.8%

       \[\leadsto 2 \cdot \color{blue}{\left(u \cdot s\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

12. Simplified 24.8%

    \[\leadsto \color{blue}{2 \cdot \left(u \cdot s\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

13. Final simplification 24.8%

    \[\leadsto 2 \cdot \left(s \cdot u\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

Alternative 7: 25.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

   2. associate-*r* 24.8%

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

   3. neg-mul-1 24.8%

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

9. Simplified 24.8%

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

10. Taylor expanded in s around 0 24.9%

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

    1. mul-1-neg 24.9%

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

    2. unsub-neg 24.9%

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

    3. log-div 24.8%

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

12. Simplified 24.8%

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

13. Final simplification 24.8%

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

Alternative 8: 25.1% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

   2. associate-*r* 24.8%

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

   3. neg-mul-1 24.8%

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

9. Simplified 24.8%

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

10. Final simplification 24.8%

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

Alternative 9: 11.6% accurate, 6.8× 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 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 10.6%

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

   1. associate--r+ 10.6%

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

   2. cancel-sign-sub-inv 10.6%

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

   3. distribute-rgt-out-- 10.6%

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

   4. *-commutative 10.6%

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

   5. metadata-eval 10.6%

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

   6. metadata-eval 10.6%

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

   7. *-commutative 10.6%

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

6. Simplified 10.6%

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

7. Taylor expanded in u around 0 10.6%

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

   1. +-commutative 10.6%

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

   2. associate-*r* 10.6%

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

   3. *-commutative 10.6%

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

   4. distribute-rgt-out 10.6%

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

   5. *-commutative 10.6%

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

9. Simplified 10.6%

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

10. Final simplification 10.6%

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

Alternative 10: 11.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

   2. associate-*r* 24.8%

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

   3. neg-mul-1 24.8%

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

9. Simplified 24.8%

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

10. Taylor expanded in s around inf 10.4%

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

    1. *-commutative 10.4%

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

    2. frac-2neg 10.4%

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

    3. associate-*l/ 10.4%

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

    4. add-sqr-sqrt -0.0%

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

    5. sqrt-unprod 8.1%

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

    6. sqr-neg 8.1%

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

    7. sqrt-unprod 4.6%

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

    8. add-sqr-sqrt 4.6%

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

    9. add-sqr-sqrt -0.0%

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

    10. sqrt-unprod 8.9%

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

    11. sqr-neg 8.9%

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

    12. sqrt-unprod 10.4%

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

    13. add-sqr-sqrt 10.4%

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

12. Applied egg-rr 10.4%

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

13. Final simplification 10.4%

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

Alternative 11: 11.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u around 0 10.4%

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

   1. neg-mul-1 10.4%

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

6. Simplified 10.4%

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

7. Final simplification 10.4%

   \[\leadsto -\pi \]

Alternative 12: 4.6% accurate, 7.3× 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 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

   2. associate-*r* 24.8%

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

   3. neg-mul-1 24.8%

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

9. Simplified 24.8%

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

    1. add-log-exp 14.3%

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

    2. *-commutative 14.3%

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

    3. log1p-udef 14.3%

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

    4. +-commutative 14.3%

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

    5. exp-to-pow 14.3%

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

    6. add-sqr-sqrt -0.0%

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

    7. sqrt-unprod 9.9%

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

    8. sqr-neg 9.9%

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

    9. sqrt-unprod 9.8%

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

    10. add-sqr-sqrt 9.8%

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

11. Applied egg-rr 9.8%

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

12. Taylor expanded in s around inf 4.6%

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

13. Final simplification 4.6%

    \[\leadsto \pi \]

Alternative 13: 10.3% accurate, 730.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.6%

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

   1. +-commutative 24.6%

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

   2. fma-def 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

   2. associate-*r* 24.8%

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

   3. neg-mul-1 24.8%

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

9. Simplified 24.8%

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

    1. add-log-exp 14.3%

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

    2. *-commutative 14.3%

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

    3. log1p-udef 14.3%

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

    4. +-commutative 14.3%

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

    5. exp-to-pow 14.3%

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

    6. add-sqr-sqrt -0.0%

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

    7. sqrt-unprod 9.9%

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

    8. sqr-neg 9.9%

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

    9. sqrt-unprod 9.8%

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

    10. add-sqr-sqrt 9.8%

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

11. Applied egg-rr 9.8%

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

12. Taylor expanded in s around 0 10.3%

    \[\leadsto \log \color{blue}{1} \]

13. Final simplification 10.3%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023279 
(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))))