Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 10.8s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 25.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(4, \left|\pi \cdot \frac{\mathsf{fma}\left(u, 0.5, -0.25\right)}{s}\right|, 1\right)\right) \cdot \left(-s\right) \end{array} \]

FPCore

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

C

float code(float u, float s) {
	return logf(fmaf(4.0f, fabsf((((float) M_PI) * (fmaf(u, 0.5f, -0.25f) / s))), 1.0f)) * -s;
}

Julia

function code(u, s)
	return Float32(log(fma(Float32(4.0), abs(Float32(Float32(pi) * Float32(fma(u, Float32(0.5), Float32(-0.25)) / s))), Float32(1.0))) * Float32(-s))
end

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around -inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-def 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

6. Simplified 24.8%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\left(\pi \cdot u\right) \cdot -0.5 + \pi \cdot 0.25}{s}, 1\right)\right)}\right) \]
7. Step-by-step derivation

   1. add-sqr-sqrt 24.8%

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

   2. sqrt-unprod 15.8%

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

   3. pow2 15.8%

      \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \sqrt{\color{blue}{{\left(\frac{\left(\pi \cdot u\right) \cdot -0.5 + \pi \cdot 0.25}{s}\right)}^{2}}}, 1\right)\right)\right) \]

8. Applied egg-rr 15.8%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(u \cdot 0.5 + -0.25\right)}{s}\right)}^{2}}}, 1\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow2 15.8%

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

   2. rem-sqrt-square 25.0%

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

   3. *-rgt-identity 25.0%

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

   4. associate-*r/ 25.0%

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

   5. associate-*l* 25.0%

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

   6. associate-*r/ 25.0%

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

   7. *-rgt-identity 25.0%

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

   8. fma-def 25.0%

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

10. Simplified 25.0%

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

11. Final simplification 25.0%

    \[\leadsto \log \left(\mathsf{fma}\left(4, \left|\pi \cdot \frac{\mathsf{fma}\left(u, 0.5, -0.25\right)}{s}\right|, 1\right)\right) \cdot \left(-s\right) \]

Alternative 3: 25.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(4, \left|\pi \cdot \left(0.5 \cdot \frac{u}{s} + 0.25 \cdot \frac{-1}{s}\right)\right|, 1\right)\right) \cdot \left(-s\right) \end{array} \]

FPCore

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

C

float code(float u, float s) {
	return logf(fmaf(4.0f, fabsf((((float) M_PI) * ((0.5f * (u / s)) + (0.25f * (-1.0f / s))))), 1.0f)) * -s;
}

Julia

function code(u, s)
	return Float32(log(fma(Float32(4.0), abs(Float32(Float32(pi) * Float32(Float32(Float32(0.5) * Float32(u / s)) + Float32(Float32(0.25) * Float32(Float32(-1.0) / s))))), Float32(1.0))) * Float32(-s))
end

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around -inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-def 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

6. Simplified 24.8%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\left(\pi \cdot u\right) \cdot -0.5 + \pi \cdot 0.25}{s}, 1\right)\right)}\right) \]
7. Step-by-step derivation

   1. add-sqr-sqrt 24.8%

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

   2. sqrt-unprod 15.8%

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

   3. pow2 15.8%

      \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \sqrt{\color{blue}{{\left(\frac{\left(\pi \cdot u\right) \cdot -0.5 + \pi \cdot 0.25}{s}\right)}^{2}}}, 1\right)\right)\right) \]

8. Applied egg-rr 15.8%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(u \cdot 0.5 + -0.25\right)}{s}\right)}^{2}}}, 1\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow2 15.8%

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

   2. rem-sqrt-square 25.0%

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

   3. *-rgt-identity 25.0%

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

   4. associate-*r/ 25.0%

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

   5. associate-*l* 25.0%

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

   6. associate-*r/ 25.0%

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

   7. *-rgt-identity 25.0%

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

   8. fma-def 25.0%

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

10. Simplified 25.0%

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

11. Taylor expanded in u around 0 25.0%

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

12. Final simplification 25.0%

    \[\leadsto \log \left(\mathsf{fma}\left(4, \left|\pi \cdot \left(0.5 \cdot \frac{u}{s} + 0.25 \cdot \frac{-1}{s}\right)\right|, 1\right)\right) \cdot \left(-s\right) \]

Alternative 4: 25.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(4 \cdot \left|\pi \cdot \left(\frac{u}{s} \cdot -0.5 + \frac{0.25}{s}\right)\right|\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (log1p (* 4.0 (fabs (* PI (+ (* (/ u s) -0.5) (/ 0.25 s)))))) (- s)))

C

float code(float u, float s) {
	return log1pf((4.0f * fabsf((((float) M_PI) * (((u / s) * -0.5f) + (0.25f / s)))))) * -s;
}

Julia

function code(u, s)
	return Float32(log1p(Float32(Float32(4.0) * abs(Float32(Float32(pi) * Float32(Float32(Float32(u / s) * Float32(-0.5)) + Float32(Float32(0.25) / s)))))) * Float32(-s))
end

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around -inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-def 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

6. Simplified 24.8%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\left(\pi \cdot u\right) \cdot -0.5 + \pi \cdot 0.25}{s}, 1\right)\right)}\right) \]
7. Step-by-step derivation

   1. add-sqr-sqrt 24.8%

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

   2. sqrt-unprod 15.8%

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

   3. pow2 15.8%

      \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \sqrt{\color{blue}{{\left(\frac{\left(\pi \cdot u\right) \cdot -0.5 + \pi \cdot 0.25}{s}\right)}^{2}}}, 1\right)\right)\right) \]

8. Applied egg-rr 15.8%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(u \cdot 0.5 + -0.25\right)}{s}\right)}^{2}}}, 1\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow2 15.8%

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

   2. rem-sqrt-square 25.0%

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

   3. *-rgt-identity 25.0%

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

   4. associate-*r/ 25.0%

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

   5. associate-*l* 25.0%

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

   6. associate-*r/ 25.0%

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

   7. *-rgt-identity 25.0%

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

   8. fma-def 25.0%

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

10. Simplified 25.0%

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

11. Taylor expanded in u around 0 25.0%

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

12. Taylor expanded in u around -inf 25.0%

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

    1. log1p-def 25.0%

       \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(4 \cdot \left|-1 \cdot \left(\left(0.25 \cdot \frac{1}{s} + -0.5 \cdot \frac{u}{s}\right) \cdot \pi\right)\right|\right)}\right) \]

    2. mul-1-neg 25.0%

       \[\leadsto s \cdot \left(-\mathsf{log1p}\left(4 \cdot \left|\color{blue}{-\left(0.25 \cdot \frac{1}{s} + -0.5 \cdot \frac{u}{s}\right) \cdot \pi}\right|\right)\right) \]

    3. *-commutative 25.0%

       \[\leadsto s \cdot \left(-\mathsf{log1p}\left(4 \cdot \left|-\color{blue}{\pi \cdot \left(0.25 \cdot \frac{1}{s} + -0.5 \cdot \frac{u}{s}\right)}\right|\right)\right) \]

    4. associate-*r/ 25.0%

       \[\leadsto s \cdot \left(-\mathsf{log1p}\left(4 \cdot \left|-\pi \cdot \left(\color{blue}{\frac{0.25 \cdot 1}{s}} + -0.5 \cdot \frac{u}{s}\right)\right|\right)\right) \]

    5. metadata-eval 25.0%

       \[\leadsto s \cdot \left(-\mathsf{log1p}\left(4 \cdot \left|-\pi \cdot \left(\frac{\color{blue}{0.25}}{s} + -0.5 \cdot \frac{u}{s}\right)\right|\right)\right) \]

    6. +-commutative 25.0%

       \[\leadsto s \cdot \left(-\mathsf{log1p}\left(4 \cdot \left|-\pi \cdot \color{blue}{\left(-0.5 \cdot \frac{u}{s} + \frac{0.25}{s}\right)}\right|\right)\right) \]

14. Simplified 25.0%

    \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(4 \cdot \left|-\pi \cdot \left(-0.5 \cdot \frac{u}{s} + \frac{0.25}{s}\right)\right|\right)}\right) \]

15. Final simplification 25.0%

    \[\leadsto \mathsf{log1p}\left(4 \cdot \left|\pi \cdot \left(\frac{u}{s} \cdot -0.5 + \frac{0.25}{s}\right)\right|\right) \cdot \left(-s\right) \]

Alternative 5: 25.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return logf((1.0f + (4.0f * fabsf((((float) M_PI) * (-0.25f / s)))))) * -s;
}

Julia

function code(u, s)
	return Float32(log(Float32(Float32(1.0) + Float32(Float32(4.0) * abs(Float32(Float32(pi) * Float32(Float32(-0.25) / s)))))) * Float32(-s))
end

MATLAB

function tmp = code(u, s)
	tmp = log((single(1.0) + (single(4.0) * abs((single(pi) * (single(-0.25) / s)))))) * -s;
end

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around -inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-def 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

6. Simplified 24.8%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\left(\pi \cdot u\right) \cdot -0.5 + \pi \cdot 0.25}{s}, 1\right)\right)}\right) \]
7. Step-by-step derivation

   1. add-sqr-sqrt 24.8%

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

   2. sqrt-unprod 15.8%

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

   3. pow2 15.8%

      \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \sqrt{\color{blue}{{\left(\frac{\left(\pi \cdot u\right) \cdot -0.5 + \pi \cdot 0.25}{s}\right)}^{2}}}, 1\right)\right)\right) \]

8. Applied egg-rr 15.8%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(u \cdot 0.5 + -0.25\right)}{s}\right)}^{2}}}, 1\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow2 15.8%

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

   2. rem-sqrt-square 25.0%

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

   3. *-rgt-identity 25.0%

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

   4. associate-*r/ 25.0%

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

   5. associate-*l* 25.0%

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

   6. associate-*r/ 25.0%

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

   7. *-rgt-identity 25.0%

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

   8. fma-def 25.0%

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

10. Simplified 25.0%

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

11. Taylor expanded in u around 0 24.9%

    \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \left|\pi \cdot \color{blue}{\frac{-0.25}{s}}\right|, 1\right)\right)\right) \]
12. Step-by-step derivation

    1. fma-udef 24.9%

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

13. Applied egg-rr 24.9%

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

14. Final simplification 24.9%

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

Alternative 6: 25.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 5.7%

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

5. Taylor expanded in s around inf 24.9%

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

   1. +-commutative 24.9%

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

7. Simplified 24.9%

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

8. Final simplification 24.9%

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

Alternative 7: 11.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 11.6%

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

   1. associate--r+ 11.6%

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

   2. cancel-sign-sub-inv 11.6%

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

   3. distribute-rgt-out-- 11.6%

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

   4. *-commutative 11.6%

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

   5. metadata-eval 11.6%

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

   6. metadata-eval 11.6%

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

   7. *-commutative 11.6%

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

6. Simplified 11.6%

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

7. Taylor expanded in u around 0 11.6%

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

   1. +-commutative 11.6%

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

   2. associate-*r* 11.6%

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

   3. *-commutative 11.6%

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

   4. distribute-rgt-in 11.6%

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

   5. fma-def 11.6%

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

9. Simplified 11.6%

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

10. Final simplification 11.6%

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

Alternative 8: 11.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{-\pi}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 5.7%

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

5. Taylor expanded in s around 0 11.2%

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

6. Final simplification 11.2%

   \[\leadsto s \cdot \frac{-\pi}{s} \]

Alternative 9: 11.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 11.2%

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

   1. mul-1-neg 11.2%

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

6. Simplified 11.2%

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

7. Final simplification 11.2%

   \[\leadsto -\pi \]

Alternative 10: 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 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 5.7%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 2.3%

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

   3. sqr-neg 2.3%

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

   4. sqrt-unprod 2.3%

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

   5. add-sqr-sqrt 2.3%

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

   6. add-log-exp 4.7%

      \[\leadsto s \cdot \color{blue}{\frac{\pi}{s}} \]

   7. clear-num 4.7%

      \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{s}{\pi}}} \]

   8. un-div-inv 4.7%

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

6. Applied egg-rr 4.7%

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

7. Taylor expanded in s around 0 4.7%

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

8. Final simplification 4.7%

   \[\leadsto \pi \]

Reproduce

herbie shell --seed 2023208 
(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))))