Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 20.1s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. u = 7.691037251511545e-10
  2. s = 0.026664163917303085

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, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\mathsf{log1p}\left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + {\left(e^{\frac{1}{s}}\right)}^{\pi}}} + -2\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log1p
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ (- PI) s))))
       (/ (- 1.0 u) (+ 1.0 (pow (exp (/ 1.0 s)) PI)))))
     -2.0)))))

C

float code(float u, float s) {
	return s * -log1pf(((1.0f / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + powf(expf((1.0f / s)), ((float) M_PI)))))) + -2.0f));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log1p(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(1.0) / s)) ^ Float32(pi)))))) + Float32(-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 98.8%

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

   1. clear-num 98.9%

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

   2. inv-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

8. Simplified 98.9%

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

   1. log1p-expm1-u 98.9%

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

   2. expm1-undefine 98.9%

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

10. Applied egg-rr 98.9%

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

    1. associate--l+ 98.9%

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

    2. metadata-eval 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

    \[\leadsto s \cdot \left(-\mathsf{log1p}\left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + {\left(e^{\frac{1}{s}}\right)}^{\pi}}} + -2\right)\right) \]
14. Add Preprocessing

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ (- PI) s))))
      (/ (- 1.0 u) (+ 1.0 (exp (/ 1.0 (/ s PI)))))))
    -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((1.0f / (s / ((float) M_PI)))))))) + -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(1.0) / Float32(s / Float32(pi)))))))) + 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(1.0) / (s / single(pi)))))))) + 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 98.8%

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

   1. clear-num 98.9%

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

   2. inv-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 3: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\mathsf{log1p}\left(-2 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log1p
    (+
     -2.0
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ (- PI) s))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))))))))

C

float code(float u, float s) {
	return s * -log1pf((-2.0f + (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(s * Float32(-log1p(Float32(Float32(-2.0) + 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))))))))))
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 98.8%

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

   1. clear-num 98.9%

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

   2. inv-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

8. Simplified 98.9%

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

   1. log1p-expm1-u 98.9%

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

   2. expm1-undefine 98.9%

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

10. Applied egg-rr 98.9%

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

    1. associate--l+ 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

    \[\leadsto s \cdot \left(-\mathsf{log1p}\left(-2 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right) \]
14. Add Preprocessing

Alternative 4: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (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(s * Float32(-log(Float32(Float32(-1.0) + 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))))))))))
end

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 5: 25.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(-log(log(Float32(Float32(exp(1)) + Float32(Float32(Float32(-4.0) * Float32(Float32(exp(1)) * Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) - fma(Float32(-0.25), Float32(u * Float32(pi)), Float32(Float32(pi) * 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 98.8%

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

   1. clear-num 98.9%

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

   2. inv-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

8. Simplified 98.9%

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

   1. add-log-exp 21.0%

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

   2. associate-/r/ 21.0%

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

   3. exp-prod 21.0%

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

10. Applied egg-rr 21.0%

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

11. Taylor expanded in s around inf 25.5%

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

    1. exp-1-e 25.5%

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

    2. associate-*r/ 25.5%

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

    3. *-commutative 25.5%

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

    4. *-commutative 25.5%

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

    5. fma-define 25.5%

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

    6. *-commutative 25.5%

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

    7. *-commutative 25.5%

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

    8. exp-1-e 25.5%

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

13. Simplified 25.5%

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

14. Final simplification 25.5%

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

Alternative 6: 25.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 25.0%

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

   1. cancel-sign-sub-inv 25.0%

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

   2. distribute-rgt-out-- 25.0%

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

   3. metadata-eval 25.0%

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

   4. metadata-eval 25.0%

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

   5. *-commutative 25.0%

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

5. Simplified 25.0%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. *-commutative 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in s around 0 25.4%

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

    1. +-commutative 25.4%

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

    2. mul-1-neg 25.4%

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

    3. unsub-neg 25.4%

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

11. Simplified 25.4%

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

12. Final simplification 25.4%

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

Alternative 7: 25.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 25.0%

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

   1. cancel-sign-sub-inv 25.0%

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

   2. distribute-rgt-out-- 25.0%

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

   3. metadata-eval 25.0%

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

   4. metadata-eval 25.0%

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

   5. *-commutative 25.0%

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

5. Simplified 25.0%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. *-commutative 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

    2. *-commutative 25.4%

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

    3. distribute-rgt-neg-in 25.4%

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

    4. associate-+r+ 25.4%

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

    5. mul-1-neg 25.4%

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

    6. unsub-neg 25.4%

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

    7. *-commutative 25.4%

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

11. Simplified 25.4%

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

12. Final simplification 25.4%

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

Alternative 8: 25.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 25.0%

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

   1. cancel-sign-sub-inv 25.0%

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

   2. distribute-rgt-out-- 25.0%

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

   3. metadata-eval 25.0%

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

   4. metadata-eval 25.0%

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

   5. *-commutative 25.0%

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

5. Simplified 25.0%

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

6. Taylor expanded in u around 0 25.2%

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

7. Final simplification 25.2%

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

Alternative 9: 25.2% accurate, 4.1× speedup

Math

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

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(s * Float32(-log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 25.0%

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

   1. cancel-sign-sub-inv 25.0%

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

   2. distribute-rgt-out-- 25.0%

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

   3. metadata-eval 25.0%

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

   4. metadata-eval 25.0%

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

   5. *-commutative 25.0%

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

5. Simplified 25.0%

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

6. Taylor expanded in u around 0 25.2%

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

   1. log1p-define 25.2%

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

8. Simplified 25.2%

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

9. Final simplification 25.2%

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

Alternative 10: 11.5% 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 98.8%

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

5. Taylor expanded in s around inf 11.7%

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

6. Final simplification 11.7%

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

Alternative 11: 11.5% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.25) * (u * single(pi))) - (single(pi) * (single(0.25) + (u * 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 98.8%

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

5. Taylor expanded in s around inf 11.7%

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

   1. *-un-lft-identity 11.7%

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

   2. fma-define 11.7%

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

   3. *-commutative 11.7%

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

7. Applied egg-rr 11.7%

   \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \color{blue}{1 \cdot \mathsf{fma}\left(-0.25, u \cdot \pi, \pi \cdot 0.25\right)}\right) \]
8. Step-by-step derivation

   1. *-lft-identity 11.7%

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

   2. fma-undefine 11.7%

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

   3. associate-*r* 11.7%

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

   4. *-commutative 11.7%

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

   5. distribute-rgt-out 11.7%

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

   6. *-commutative 11.7%

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

9. Simplified 11.7%

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

10. Final simplification 11.7%

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

Alternative 12: 11.5% 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%

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

3. Taylor expanded in s around inf 11.7%

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

   1. cancel-sign-sub-inv 11.7%

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

   2. metadata-eval 11.7%

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

   3. distribute-rgt-out-- 11.7%

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

   4. metadata-eval 11.7%

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

   5. *-commutative 11.7%

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

5. Simplified 11.7%

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

6. Taylor expanded in u around inf 11.7%

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

   1. +-commutative 11.7%

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

   2. mul-1-neg 11.7%

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

   3. unsub-neg 11.7%

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

   4. *-commutative 11.7%

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

8. Simplified 11.7%

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

Alternative 13: 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%

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

3. Taylor expanded in s around inf 11.7%

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

   1. cancel-sign-sub-inv 11.7%

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

   2. metadata-eval 11.7%

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

   3. distribute-rgt-out-- 11.7%

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

   4. metadata-eval 11.7%

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

   5. *-commutative 11.7%

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

5. Simplified 11.7%

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

6. Taylor expanded in u around 0 11.7%

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

   1. neg-mul-1 11.7%

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

   2. +-commutative 11.7%

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

   3. associate-*r* 11.7%

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

   4. neg-mul-1 11.7%

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

   5. distribute-rgt-out 11.7%

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

8. Simplified 11.7%

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

9. Final simplification 11.7%

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

Alternative 14: 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 98.8%

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

5. Taylor expanded in u around 0 11.5%

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

   1. mul-1-neg 11.5%

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

7. Simplified 11.5%

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

Alternative 15: 10.3% accurate, 433.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 10.7%

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

6. Taylor expanded in s around 0 10.7%

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

Reproduce

herbie shell --seed 2024180 
(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))))