Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 18.5s

Alternatives: 12

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. add-sqr-sqrt 98.9%

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

   3. times-frac 98.9%

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

6. Applied egg-rr 98.9%

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

   1. associate-*l/ 98.9%

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

   2. *-lft-identity 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (pow E (/ 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 + powf(((float) M_E), (((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) + (Float32(exp(1)) ^ Float32(Float32(pi) / s)))))))))
end

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. exp-prod 98.9%

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 98.9% 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(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 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 4: 92.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + \left(1 + \frac{\pi - -0.5 \cdot \frac{\pi \cdot \pi}{s}}{s}\right)}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (+ 1.0 (/ (- PI (* -0.5 (/ (* PI PI) s))) 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 + (1.0f + ((((float) M_PI) - (-0.5f * ((((float) M_PI) * ((float) M_PI)) / s))) / 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) + Float32(Float32(1.0) + Float32(Float32(Float32(pi) - Float32(Float32(-0.5) * Float32(Float32(Float32(pi) * Float32(pi)) / s))) / 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) + (single(1.0) + ((single(pi) - (single(-0.5) * ((single(pi) * single(pi)) / s))) / s))))))));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 90.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \pi + -0.5 \cdot \frac{{\pi}^{2}}{s}}{s}\right)}}} + -1\right) \]
6. Step-by-step derivation

   1. unpow2 90.4%

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

7. Applied egg-rr 90.4%

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

8. Final simplification 90.4%

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

Alternative 5: 85.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (+ 1.0 (/ 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 + (1.0f + (((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) + Float32(Float32(1.0) + Float32(Float32(pi) / s)))))))))
end

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 83.8%

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

   1. +-commutative 83.8%

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

7. Simplified 83.8%

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

8. Final simplification 83.8%

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

Alternative 6: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{\frac{-1}{u}}{0.5 + \frac{1}{-1 - e^{\frac{\pi}{-s}}}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log (+ -1.0 (/ (/ -1.0 u) (+ 0.5 (/ 1.0 (- -1.0 (exp (/ PI (- s)))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + ((-1.0f / u) / (0.5f + (1.0f / (-1.0f - expf((((float) M_PI) / -s))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(Float32(-1.0) / u) / Float32(Float32(0.5) + Float32(Float32(1.0) / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / Float32(-s))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + ((single(-1.0) / u) / (single(0.5) + (single(1.0) / (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 98.9%

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

5. Taylor expanded in s around inf 8.8%

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

6. Taylor expanded in u around -inf 37.2%

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

   1. associate-/r* 37.2%

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

   2. mul-1-neg 37.2%

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

   3. distribute-frac-neg2 37.2%

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

8. Simplified 37.2%

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

9. Final simplification 37.2%

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

Alternative 7: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{\frac{1}{\frac{1}{1 + e^{\frac{\pi}{-s}}} - 0.5} - u}{u}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log (/ (- (/ 1.0 (- (/ 1.0 (+ 1.0 (exp (/ PI (- s))))) 0.5)) u) u))))

C

float code(float u, float s) {
	return -s * logf((((1.0f / ((1.0f / (1.0f + expf((((float) M_PI) / -s)))) - 0.5f)) - u) / u));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) - Float32(0.5))) - u) / u)))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((((single(1.0) / ((single(1.0) / (single(1.0) + exp((single(pi) / -s)))) - single(0.5))) - u) / u));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 8.8%

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

6. Taylor expanded in u around -inf 37.2%

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

   1. associate-/r* 37.2%

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

   2. mul-1-neg 37.2%

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

   3. distribute-frac-neg2 37.2%

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

8. Simplified 37.2%

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

9. Taylor expanded in u around 0 37.2%

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

10. Final simplification 37.2%

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

Alternative 8: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(\frac{\frac{-1}{u}}{0.5 + \frac{1}{-1 - e^{\frac{\pi}{-s}}}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (- (log (/ (/ -1.0 u) (+ 0.5 (/ 1.0 (- -1.0 (exp (/ PI (- s)))))))))))

C

float code(float u, float s) {
	return s * -logf(((-1.0f / u) / (0.5f + (1.0f / (-1.0f - expf((((float) M_PI) / -s)))))));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(Float32(-1.0) / u) / Float32(Float32(0.5) + Float32(Float32(1.0) / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / Float32(-s))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(-1.0) / u) / (single(0.5) + (single(1.0) / (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 98.9%

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

5. Taylor expanded in s around inf 8.8%

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

6. Taylor expanded in u around -inf 37.2%

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

   1. associate-/r* 37.2%

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

   2. mul-1-neg 37.2%

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

   3. distribute-frac-neg2 37.2%

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

8. Simplified 37.2%

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

9. Taylor expanded in u around 0 37.2%

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

    1. associate-/r* 37.2%

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

    2. mul-1-neg 37.2%

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

    3. distribute-frac-neg2 37.2%

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

    4. distribute-frac-neg2 37.2%

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

    5. mul-1-neg 37.2%

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

    6. associate-*r/ 37.2%

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

    7. mul-1-neg 37.2%

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

11. Simplified 37.2%

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

12. Final simplification 37.2%

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

Alternative 9: 11.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-4.0) * Float32((cbrt(Float32(pi)) ^ Float32(3.0)) * Float32(Float32(0.25) + Float32(u * Float32(-0.5)))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 12.4%

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

   1. associate--r+ 12.4%

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

   2. cancel-sign-sub-inv 12.4%

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

   3. cancel-sign-sub-inv 12.4%

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

   4. metadata-eval 12.4%

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

   5. associate-*r* 12.4%

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

   6. distribute-rgt-out 12.4%

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

   7. *-commutative 12.4%

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

   8. metadata-eval 12.4%

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

   9. *-commutative 12.4%

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

   10. associate-*l* 12.4%

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

7. Simplified 12.4%

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

   1. add-cube-cbrt 12.4%

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

   2. pow3 12.4%

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

9. Applied egg-rr 12.4%

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

10. Taylor expanded in u around 0 12.4%

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

    1. +-commutative 12.4%

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

    2. associate-*r* 12.4%

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

    3. distribute-rgt-out 12.4%

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

12. Simplified 12.4%

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

    1. add-cube-cbrt 12.4%

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

    2. pow3 12.4%

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

14. Applied egg-rr 12.4%

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

15. Final simplification 12.4%

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

Alternative 10: 11.7% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 12.4%

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

   1. associate--r+ 12.4%

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

   2. cancel-sign-sub-inv 12.4%

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

   3. cancel-sign-sub-inv 12.4%

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

   4. metadata-eval 12.4%

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

   5. associate-*r* 12.4%

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

   6. distribute-rgt-out 12.4%

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

   7. *-commutative 12.4%

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

   8. metadata-eval 12.4%

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

   9. *-commutative 12.4%

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

   10. associate-*l* 12.4%

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

7. Simplified 12.4%

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

8. Taylor expanded in u around inf 12.4%

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

9. Final simplification 12.4%

   \[\leadsto -4 \cdot \left(u \cdot \left(\pi \cdot -0.5 + 0.25 \cdot \frac{\pi}{u}\right)\right) \]
10. Add Preprocessing

Alternative 11: 11.7% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 12.4%

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

   1. associate--r+ 12.4%

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

   2. cancel-sign-sub-inv 12.4%

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

   3. cancel-sign-sub-inv 12.4%

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

   4. metadata-eval 12.4%

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

   5. associate-*r* 12.4%

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

   6. distribute-rgt-out 12.4%

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

   7. *-commutative 12.4%

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

   8. metadata-eval 12.4%

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

   9. *-commutative 12.4%

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

   10. associate-*l* 12.4%

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

7. Simplified 12.4%

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

   1. add-cube-cbrt 12.4%

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

   2. pow3 12.4%

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

9. Applied egg-rr 12.4%

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

10. Taylor expanded in u around 0 12.4%

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

    1. +-commutative 12.4%

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

    2. associate-*r* 12.4%

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

    3. distribute-rgt-out 12.4%

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

12. Simplified 12.4%

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

13. Final simplification 12.4%

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

Alternative 12: 11.5% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 12.2%

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

   1. neg-mul-1 12.2%

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

7. Simplified 12.2%

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

8. Final simplification 12.2%

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

Reproduce

herbie shell --seed 2024067 
(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))))