Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 16.4s

Alternatives: 13

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ (- PI) s))))
      (/ (- 1.0 u) (+ 1.0 (pow E (* PI (/ 1.0 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 + powf(((float) M_E), (((float) M_PI) * (1.0f / s))))))) + -1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(pi) * Float32(Float32(1.0) / 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) + (single(2.71828182845904523536) ^ (single(pi) * (single(1.0) / s))))))) + single(-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. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.0%

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

   2. exp-prod 99.0%

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

6. Applied egg-rr 99.0%

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

   1. clear-num 99.0%

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

   2. associate-/r/ 99.1%

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

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

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(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) + (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 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. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.0%

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

   2. exp-prod 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 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 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. Add Preprocessing

5. Final simplification 99.0%

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

Alternative 4: 85.9% 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(-Float32(pi)) / 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 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. Add Preprocessing

5. Taylor expanded in s around inf 88.2%

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

   1. +-commutative 88.2%

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

7. Simplified 88.2%

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

8. Final simplification 88.2%

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

Alternative 5: 37.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Taylor expanded in s around inf 8.9%

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

6. Taylor expanded in u around inf 37.3%

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

   1. sub-neg 37.3%

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

   2. sub-neg 37.3%

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

   3. associate-*r/ 37.3%

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

   4. mul-1-neg 37.3%

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

   5. metadata-eval 37.3%

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

   6. metadata-eval 37.3%

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

8. Simplified 37.3%

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

9. Final simplification 37.3%

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

Alternative 6: 37.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Taylor expanded in s around inf 8.9%

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

6. Taylor expanded in u around inf 37.3%

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

   1. sub-neg 37.3%

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

   2. sub-neg 37.3%

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

   3. associate-*r/ 37.3%

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

   4. mul-1-neg 37.3%

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

   5. metadata-eval 37.3%

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

   6. metadata-eval 37.3%

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

8. Simplified 37.3%

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

9. Taylor expanded in u around 0 37.3%

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

    1. associate-/r* 37.3%

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

    2. sub-neg 37.3%

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

    3. neg-mul-1 37.3%

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

    4. distribute-frac-neg2 37.3%

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

    5. metadata-eval 37.3%

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

    6. +-commutative 37.3%

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

    7. distribute-frac-neg2 37.3%

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

    8. neg-mul-1 37.3%

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

    9. neg-mul-1 37.3%

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

    10. distribute-neg-frac 37.3%

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

11. Simplified 37.3%

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

12. Final simplification 37.3%

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

Alternative 7: 16.2% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(-4 \cdot \frac{1}{-1 + \left(-1 + \frac{\pi}{s}\right)} - 2\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (* u (- (* -4.0 (/ 1.0 (+ -1.0 (+ -1.0 (/ PI s))))) 2.0))))

C

float code(float u, float s) {
	return s * (u * ((-4.0f * (1.0f / (-1.0f + (-1.0f + (((float) M_PI) / s))))) - 2.0f));
}

Julia

function code(u, s)
	return Float32(s * Float32(u * Float32(Float32(Float32(-4.0) * Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(-1.0) + Float32(Float32(pi) / s))))) - Float32(2.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (u * ((single(-4.0) * (single(1.0) / (single(-1.0) + (single(-1.0) + (single(pi) / s))))) - single(2.0)));
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. Add Preprocessing

5. Taylor expanded in s around inf 8.9%

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

6. Taylor expanded in u around 0 8.7%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \left(u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - 0.5\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 8.7%

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

   2. associate-*r* 8.7%

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

   3. *-commutative 8.7%

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

   4. sub-neg 8.7%

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

   5. metadata-eval 8.7%

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

   6. distribute-lft-in 8.7%

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

   7. mul-1-neg 8.7%

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

   8. distribute-neg-frac2 8.7%

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

   9. metadata-eval 8.7%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot \left(-4 \cdot \frac{1}{1 + e^{\frac{\pi}{-s}}} + \color{blue}{2}\right)\right) \]

8. Simplified 8.7%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(u \cdot \left(-4 \cdot \frac{1}{1 + e^{\frac{\pi}{-s}}} + 2\right)\right)} \]

9. Taylor expanded in s around inf 16.4%

   \[\leadsto \left(-s\right) \cdot \left(u \cdot \left(-4 \cdot \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{\pi}{s}\right)}} + 2\right)\right) \]
10. Step-by-step derivation

    1. mul-1-neg 16.4%

       \[\leadsto \left(-s\right) \cdot \left(u \cdot \left(-4 \cdot \frac{1}{1 + \left(1 + \color{blue}{\left(-\frac{\pi}{s}\right)}\right)} + 2\right)\right) \]

    2. unsub-neg 16.4%

       \[\leadsto \left(-s\right) \cdot \left(u \cdot \left(-4 \cdot \frac{1}{1 + \color{blue}{\left(1 - \frac{\pi}{s}\right)}} + 2\right)\right) \]

11. Simplified 16.4%

    \[\leadsto \left(-s\right) \cdot \left(u \cdot \left(-4 \cdot \frac{1}{1 + \color{blue}{\left(1 - \frac{\pi}{s}\right)}} + 2\right)\right) \]

12. Final simplification 16.4%

    \[\leadsto s \cdot \left(u \cdot \left(-4 \cdot \frac{1}{-1 + \left(-1 + \frac{\pi}{s}\right)} - 2\right)\right) \]
13. Add Preprocessing

Alternative 8: 11.7% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(u \cdot \left(\pi \cdot -0.5 + 0.25 \cdot \frac{1}{\frac{1}{\frac{\pi}{u}}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* -4.0 (* u (+ (* PI -0.5) (* 0.25 (/ 1.0 (/ 1.0 (/ PI u))))))))

C

float code(float u, float s) {
	return -4.0f * (u * ((((float) M_PI) * -0.5f) + (0.25f * (1.0f / (1.0f / (((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(1.0) / Float32(Float32(1.0) / 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(1.0) / (single(1.0) / (single(pi) / u))))));
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. Add Preprocessing

5. Taylor expanded in s around -inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. cancel-sign-sub-inv 11.4%

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

   4. metadata-eval 11.4%

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

   5. associate-*r* 11.4%

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

   6. distribute-rgt-out 11.4%

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

   7. *-commutative 11.4%

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

   8. metadata-eval 11.4%

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

   9. *-commutative 11.4%

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

   10. associate-*l* 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in u around inf 11.4%

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

   1. clear-num 11.4%

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

   2. inv-pow 11.4%

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

10. Applied egg-rr 11.4%

    \[\leadsto -4 \cdot \left(u \cdot \left(-0.5 \cdot \pi + 0.25 \cdot \color{blue}{{\left(\frac{u}{\pi}\right)}^{-1}}\right)\right) \]
11. Step-by-step derivation

    1. unpow-1 11.4%

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

12. Simplified 11.4%

    \[\leadsto -4 \cdot \left(u \cdot \left(-0.5 \cdot \pi + 0.25 \cdot \color{blue}{\frac{1}{\frac{u}{\pi}}}\right)\right) \]
13. Step-by-step derivation

    1. clear-num 11.4%

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

    2. inv-pow 11.4%

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

14. Applied egg-rr 11.4%

    \[\leadsto -4 \cdot \left(u \cdot \left(-0.5 \cdot \pi + 0.25 \cdot \frac{1}{\color{blue}{{\left(\frac{\pi}{u}\right)}^{-1}}}\right)\right) \]
15. Step-by-step derivation

    1. unpow-1 11.4%

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

16. Simplified 11.4%

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

17. Final simplification 11.4%

    \[\leadsto -4 \cdot \left(u \cdot \left(\pi \cdot -0.5 + 0.25 \cdot \frac{1}{\frac{1}{\frac{\pi}{u}}}\right)\right) \]
18. Add Preprocessing

Alternative 9: 11.7% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Taylor expanded in s around -inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. cancel-sign-sub-inv 11.4%

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

   4. metadata-eval 11.4%

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

   5. associate-*r* 11.4%

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

   6. distribute-rgt-out 11.4%

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

   7. *-commutative 11.4%

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

   8. metadata-eval 11.4%

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

   9. *-commutative 11.4%

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

   10. associate-*l* 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in u around inf 11.4%

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

9. Taylor expanded in u around 0 11.4%

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

    1. +-commutative 11.4%

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

    2. associate-*r* 11.4%

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

    3. *-commutative 11.4%

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

    4. distribute-rgt-out 11.4%

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

11. Simplified 11.4%

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

5. Taylor expanded in s around -inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. cancel-sign-sub-inv 11.4%

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

   4. metadata-eval 11.4%

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

   5. associate-*r* 11.4%

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

   6. distribute-rgt-out 11.4%

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

   7. *-commutative 11.4%

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

   8. metadata-eval 11.4%

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

   9. *-commutative 11.4%

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

   10. associate-*l* 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in u around inf 11.4%

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

9. Final simplification 11.4%

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

5. Taylor expanded in s around -inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. cancel-sign-sub-inv 11.4%

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

   4. metadata-eval 11.4%

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

   5. associate-*r* 11.4%

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

   6. distribute-rgt-out 11.4%

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

   7. *-commutative 11.4%

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

   8. metadata-eval 11.4%

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

   9. *-commutative 11.4%

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

   10. associate-*l* 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in u around 0 11.4%

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

   1. associate-*r* 11.4%

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

   2. distribute-rgt-out 11.4%

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

10. Simplified 11.4%

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

11. Final simplification 11.4%

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

Alternative 12: 11.5% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * (u * (single(0.25) * (single(pi) / u)));
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. Add Preprocessing

5. Taylor expanded in s around -inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. cancel-sign-sub-inv 11.4%

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

   4. metadata-eval 11.4%

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

   5. associate-*r* 11.4%

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

   6. distribute-rgt-out 11.4%

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

   7. *-commutative 11.4%

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

   8. metadata-eval 11.4%

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

   9. *-commutative 11.4%

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

   10. associate-*l* 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in u around inf 11.4%

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

9. Taylor expanded in u around 0 11.3%

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

Alternative 13: 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 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. Add Preprocessing

5. Taylor expanded in u around 0 11.3%

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

   1. neg-mul-1 11.3%

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

7. Simplified 11.3%

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

Reproduce

herbie shell --seed 2024085 
(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))))