Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 17.1s

Alternatives: 9

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (exp (* PI (/ 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 + expf((((float) M_PI) * (1.0f / s))))))) + -1.0f));
}

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. clear-num 98.9%

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

   2. associate-/r/ 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.4× 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%

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

4. Final simplification 98.9%

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

Alternative 3: 37.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u / 2.0f) + ((1.0f - u) / (1.0f + expf(expf(logf((((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(2.0)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(exp(log(Float32(Float32(pi) / s))))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u / single(2.0)) + ((single(1.0) - u) / (single(1.0) + exp(exp(log((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%

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

4. Taylor expanded in s around inf 37.6%

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

   1. add-exp-log 37.6%

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

6. Applied egg-rr 37.6%

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

7. Final simplification 37.6%

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

Alternative 4: 37.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u / single(2.0)) + ((single(1.0) - u) / (single(1.0) + exp(((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 98.9%

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

4. Taylor expanded in s around inf 37.6%

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

   1. clear-num 37.6%

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

   2. inv-pow 37.6%

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

6. Applied egg-rr 37.6%

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

7. Final simplification 37.6%

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

Alternative 5: 37.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around inf 37.6%

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

5. Final simplification 37.6%

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

Alternative 6: 36.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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

4. Taylor expanded in s around inf 37.6%

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

5. Taylor expanded in s around inf 36.0%

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

   1. +-commutative 36.0%

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

7. Simplified 36.0%

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

8. Final simplification 36.0%

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

Alternative 7: 11.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around inf 11.9%

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

   1. associate--r+ 11.9%

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

   2. cancel-sign-sub-inv 11.9%

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

   3. distribute-rgt-out-- 11.9%

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

   4. *-commutative 11.9%

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

   5. metadata-eval 11.9%

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

   6. metadata-eval 11.9%

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

   7. *-commutative 11.9%

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

6. Simplified 11.9%

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

7. Taylor expanded in u around 0 11.9%

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

   1. +-commutative 11.9%

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

   2. associate-*r* 11.9%

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

   3. *-commutative 11.9%

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

   4. *-commutative 11.9%

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

   5. *-commutative 11.9%

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

   6. fma-udef 11.9%

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

9. Simplified 11.9%

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

10. Taylor expanded in u around 0 11.9%

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

    1. *-commutative 11.9%

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

    2. associate-*r* 11.9%

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

    3. *-commutative 11.9%

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

    4. *-commutative 11.9%

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

    5. +-commutative 11.9%

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

    6. distribute-lft-out 11.9%

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

12. Simplified 11.9%

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

13. Final simplification 11.9%

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

Alternative 8: 11.3% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

6. Simplified 11.6%

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

7. Final simplification 11.6%

   \[\leadsto -\pi \]

Alternative 9: 4.6% accurate, 7.3× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 PI)

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around inf 37.6%

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

   1. clear-num 37.6%

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

   2. inv-pow 37.6%

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

6. Applied egg-rr 37.6%

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

8. Applied egg-rr 10.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot \log \left(-1 + \frac{1}{\mathsf{fma}\left(u, 0.5, \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 7.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \log \left(-1 + \frac{1}{\mathsf{fma}\left(u, 0.5, \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}\right)\right)\right)} \]

   2. expm1-log1p 7.9%

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

   3. log-pow 10.1%

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

10. Simplified 10.1%

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

11. Taylor expanded in u around 0 4.7%

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

12. Final simplification 4.7%

    \[\leadsto \pi \]

Reproduce

herbie shell --seed 2023331 
(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))))