Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 17.5s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(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) / s)))))) + Float32(-2.0))))
end

Derivation

1. Initial program 99.1%

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

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. add-exp-log 99.0%

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

6. Applied egg-rr 99.0%

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

   1. log1p-expm1-u 99.0%

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

   2. expm1-undefine 99.0%

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

8. Applied egg-rr 99.1%

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

   1. associate--l+ 99.1%

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

   2. metadata-eval 99.1%

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

10. Simplified 99.1%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.1%

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

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. Add Preprocessing

Alternative 3: 86.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

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 86.1%

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

6. Final simplification 86.1%

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

Alternative 4: 25.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-define 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in s around 0 25.0%

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

9. Taylor expanded in u around 0 25.2%

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

10. Final simplification 25.2%

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

Alternative 5: 25.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(Float32(2.0) * Float32(u * Float32(Float32(pi) / Float32(Float32(1.0) + Float32(Float32(pi) / s))))) - Float32(s * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-define 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in u around 0 25.0%

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

   1. +-commutative 25.0%

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

   2. mul-1-neg 25.0%

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

   3. unsub-neg 25.0%

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

   4. associate-/l* 25.0%

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

   5. log1p-define 25.0%

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

10. Simplified 25.0%

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

Alternative 6: 25.1% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-define 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in u around 0 25.0%

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

   1. associate-*r* 25.0%

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

   2. neg-mul-1 25.0%

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

   3. log1p-define 25.0%

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

10. Simplified 25.0%

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

Alternative 7: 13.0% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. metadata-eval 11.8%

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

   4. cancel-sign-sub-inv 11.8%

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

   5. associate-*r* 11.8%

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

   6. distribute-rgt-out 11.8%

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

   7. metadata-eval 11.8%

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

   8. associate-*r* 11.8%

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

7. Simplified 11.8%

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

   1. add-sqr-sqrt 11.8%

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

   2. distribute-rgt-neg-in 11.8%

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

9. Applied egg-rr 11.8%

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

    1. distribute-rgt-neg-out 11.8%

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

    2. add-sqr-sqrt 11.8%

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

    3. expm1-log1p-u 11.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-s\right)\right)} \cdot \left(4 \cdot \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \left(-0.25 \cdot u\right) \cdot \pi}{s}\right) \]

    4. expm1-undefine 13.2%

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

11. Applied egg-rr 13.2%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-s\right)} - 1\right)} \cdot \left(4 \cdot \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \left(-0.25 \cdot u\right) \cdot \pi}{s}\right) \]
12. Step-by-step derivation

    1. sub-neg 13.2%

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

    2. log1p-undefine 13.2%

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

    3. rem-exp-log 13.2%

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

    4. unsub-neg 13.2%

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

    5. metadata-eval 13.2%

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

13. Simplified 13.2%

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

14. Final simplification 13.2%

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

Alternative 8: 11.3% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 11.9%

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

   1. associate--r+ 11.9%

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

   2. cancel-sign-sub-inv 11.9%

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

   3. metadata-eval 11.9%

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

   4. cancel-sign-sub-inv 11.9%

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

   5. associate-*r* 11.9%

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

   6. distribute-rgt-out 11.9%

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

   7. metadata-eval 11.9%

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

   8. associate-*r* 11.9%

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

7. Simplified 11.9%

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

8. Taylor expanded in u around 0 11.9%

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

   1. neg-mul-1 11.9%

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

   2. +-commutative 11.9%

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

   3. unsub-neg 11.9%

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

   4. associate-*r* 11.9%

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

10. Simplified 11.9%

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

11. Final simplification 11.9%

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

Alternative 9: 11.1% 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 11.7%

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

   1. neg-mul-1 11.7%

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

7. Simplified 11.7%

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

Alternative 10: 10.4% accurate, 433.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 10.3%

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

6. Taylor expanded in s around 0 10.3%

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

Reproduce

herbie shell --seed 2024172 
(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))))