Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 19.2s

Alternatives: 12

Speedup: 0.8×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(1.0) / Float32(Float32(-1.0) - t_0))), Float32(Float32(1.0) / Float32(Float32(1.0) + t_0)))) + Float32(-1.0))))
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. sub-neg 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

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(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) + 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 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * log1pf(((1.0f / ((1.0f / (1.0f + expf((((float) M_PI) / s)))) + (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(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) + Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))))) + Float32(-2.0))))
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. sub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. unsub-neg 96.3%

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

   3. +-commutative 96.3%

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

   4. mul-1-neg 96.3%

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

   5. unsub-neg 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in s around 0 98.4%

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

   1. associate-*r* 98.4%

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

   2. neg-mul-1 98.4%

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

   3. sub-neg 98.4%

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

   4. mul-1-neg 98.4%

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

   5. metadata-eval 98.4%

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

10. Simplified 98.4%

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

    1. log1p-expm1-u 98.4%

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

    2. expm1-undefine 98.4%

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

    3. add-exp-log 98.4%

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

    4. distribute-neg-frac 98.4%

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

12. Applied egg-rr 98.4%

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

    1. associate--l+ 98.4%

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

14. Simplified 98.4%

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

Alternative 4: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(1.0) / ((single(1.0) / (single(1.0) + exp((single(pi) / s)))) + (u / (single(1.0) + exp((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. sub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. unsub-neg 96.3%

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

   3. +-commutative 96.3%

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

   4. mul-1-neg 96.3%

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

   5. unsub-neg 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in s around 0 98.4%

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

   1. associate-*r* 98.4%

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

   2. neg-mul-1 98.4%

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

   3. sub-neg 98.4%

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

   4. mul-1-neg 98.4%

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

   5. metadata-eval 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 5: 97.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + ((single(1.0) / u) + (exp((single(pi) / -s)) / u))));
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. sub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. unsub-neg 96.3%

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

   3. +-commutative 96.3%

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

   4. mul-1-neg 96.3%

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

   5. unsub-neg 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in s around 0 98.4%

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

   1. associate-*r* 98.4%

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

   2. neg-mul-1 98.4%

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

   3. sub-neg 98.4%

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

   4. mul-1-neg 98.4%

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

   5. metadata-eval 98.4%

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

10. Simplified 98.4%

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

11. Taylor expanded in u around inf 98.3%

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

12. Final simplification 98.3%

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

Alternative 6: 25.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(Float32(2.0) * Float32(s * u)) - Float32(s * log1p(Float32(Float32(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. sub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 25.1%

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

   1. associate-*r/ 25.1%

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

   2. cancel-sign-sub-inv 25.1%

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

   3. distribute-rgt-out-- 25.1%

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

   4. metadata-eval 25.1%

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

   5. metadata-eval 25.1%

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

   6. *-commutative 25.1%

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

7. Simplified 25.1%

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

8. Taylor expanded in u around 0 25.5%

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

   1. +-commutative 25.5%

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

   2. mul-1-neg 25.5%

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

   3. unsub-neg 25.5%

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

   4. associate-/l* 25.5%

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

   5. log1p-define 25.5%

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

10. Simplified 25.5%

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

11. Taylor expanded in s around 0 25.6%

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

12. Final simplification 25.6%

    \[\leadsto 2 \cdot \left(s \cdot u\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
13. Add Preprocessing

Alternative 7: 25.2% accurate, 4.1× speedup

Math

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

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(s * Float32(-log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.5%

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

   1. associate-*r* 25.5%

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

   2. neg-mul-1 25.5%

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

   3. log1p-define 25.5%

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

8. Simplified 25.5%

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

9. Final simplification 25.5%

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

Alternative 8: 11.7% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = u * ((single(pi) * single(2.0)) - (single(pi) / u));
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. sub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 11.5%

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

   1. associate-*r/ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. distribute-rgt-out-- 11.5%

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

   4. metadata-eval 11.5%

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

   5. metadata-eval 11.5%

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

   6. *-commutative 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in u around inf 11.5%

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

   1. +-commutative 11.5%

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

   2. mul-1-neg 11.5%

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

   3. unsub-neg 11.5%

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

10. Simplified 11.5%

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

11. Final simplification 11.5%

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

Alternative 9: 11.7% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (u * single(pi))) - 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. sub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 11.5%

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

   1. associate-*r/ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. distribute-rgt-out-- 11.5%

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

   4. metadata-eval 11.5%

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

   5. metadata-eval 11.5%

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

   6. *-commutative 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in u around 0 11.5%

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

   1. neg-mul-1 11.5%

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

   2. +-commutative 11.5%

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

   3. unsub-neg 11.5%

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

   4. *-commutative 11.5%

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

10. Simplified 11.5%

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

11. Final simplification 11.5%

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

Alternative 10: 11.5% accurate, 72.2× 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(pi) / Float32(-s)))
end

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 10.9%

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

6. Final simplification 10.9%

   \[\leadsto s \cdot \frac{\pi}{-s} \]
7. Add Preprocessing

Alternative 11: 11.5% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 10.9%

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

   1. mul-1-neg 10.9%

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

7. Simplified 10.9%

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

Alternative 12: 10.3% 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 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. sub-neg 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 10.0%

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

6. Taylor expanded in s around 0 10.0%

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

Reproduce

herbie shell --seed 2024137 
(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))))