Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 13.7s

Alternatives: 13

Speedup: 1.0×

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, 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

Derivation

1. Initial program 98.9%

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

Alternative 2: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.8

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

4. Applied rewrites 94.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 85.4

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

7. Applied rewrites 85.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 85.4

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

9. Applied rewrites 85.4%

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

10. Taylor expanded in u around inf

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

    1. lower-*.f32 N/A

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

    2. lower--.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-+.f32 N/A

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

    5. lower-exp.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. lower-PI.f32 N/A

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

    9. lower-/.f32 N/A

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

12. Applied rewrites 97.5%

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

Alternative 3: 85.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{\pi}{s} - -2}\\ \log \left(\frac{1}{\mathsf{fma}\left(\frac{-1}{-1 - e^{\frac{-\pi}{s}}} - t\_0, u, t\_0\right)} - 1\right) \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (- (/ PI s) -2.0))))
   (*
    (log
     (- (/ 1.0 (fma (- (/ -1.0 (- -1.0 (exp (/ (- PI) s)))) t_0) u t_0)) 1.0))
    (- s))))

C

float code(float u, float s) {
	float t_0 = 1.0f / ((((float) M_PI) / s) - -2.0f);
	return logf(((1.0f / fmaf(((-1.0f / (-1.0f - expf((-((float) M_PI) / s)))) - t_0), u, t_0)) - 1.0f)) * -s;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(Float32(pi) / s) - Float32(-2.0)))
	return Float32(log(Float32(Float32(Float32(1.0) / fma(Float32(Float32(Float32(-1.0) / Float32(Float32(-1.0) - exp(Float32(Float32(-Float32(pi)) / s)))) - t_0), u, t_0)) - Float32(1.0))) * Float32(-s))
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.8

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

4. Applied rewrites 94.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 85.4

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

7. Applied rewrites 85.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 85.4

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

9. Applied rewrites 85.4%

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

Alternative 4: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \frac{\pi}{s}\\ \left(-\log \left(\frac{1}{\frac{\mathsf{fma}\left(u, \frac{t\_0}{e^{\frac{-\pi}{s}} - -1} - 1, 1\right)}{t\_0}} - 1\right)\right) \cdot s \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 2.0 (/ PI s))))
   (*
    (-
     (log
      (-
       (/ 1.0 (/ (fma u (- (/ t_0 (- (exp (/ (- PI) s)) -1.0)) 1.0) 1.0) t_0))
       1.0)))
    s)))

C

float code(float u, float s) {
	float t_0 = 2.0f + (((float) M_PI) / s);
	return -logf(((1.0f / (fmaf(u, ((t_0 / (expf((-((float) M_PI) / s)) - -1.0f)) - 1.0f), 1.0f) / t_0)) - 1.0f)) * s;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(2.0) + Float32(Float32(pi) / s))
	return Float32(Float32(-log(Float32(Float32(Float32(1.0) / Float32(fma(u, Float32(Float32(t_0 / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0))) - Float32(1.0)), Float32(1.0)) / t_0)) - Float32(1.0)))) * s)
end

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 3.9%

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

4. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 5.9

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

6. Applied rewrites 5.9%

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

7. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 84.9

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

9. Applied rewrites 84.9%

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

Alternative 5: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\left(1 + 2 \cdot \frac{u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}{s}\right) \cdot \frac{1}{\frac{\pi}{s} - -2}} - 1\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     1.0
     (*
      (+ 1.0 (* 2.0 (/ (* u (- (* 0.25 PI) (* -0.25 PI))) s)))
      (/ 1.0 (- (/ PI s) -2.0))))
    1.0))
  (- s)))

C

float code(float u, float s) {
	return logf(((1.0f / ((1.0f + (2.0f * ((u * ((0.25f * ((float) M_PI)) - (-0.25f * ((float) M_PI)))) / s))) * (1.0f / ((((float) M_PI) / s) - -2.0f)))) - 1.0f)) * -s;
}

Julia

function code(u, s)
	return Float32(log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + Float32(Float32(2.0) * Float32(Float32(u * Float32(Float32(Float32(0.25) * Float32(pi)) - Float32(Float32(-0.25) * Float32(pi)))) / s))) * Float32(Float32(1.0) / Float32(Float32(Float32(pi) / s) - Float32(-2.0))))) - Float32(1.0))) * Float32(-s))
end

MATLAB

function tmp = code(u, s)
	tmp = log(((single(1.0) / ((single(1.0) + (single(2.0) * ((u * ((single(0.25) * single(pi)) - (single(-0.25) * single(pi)))) / s))) * (single(1.0) / ((single(pi) / s) - single(-2.0))))) - single(1.0))) * -s;
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.8

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

4. Applied rewrites 94.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 85.4

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

7. Applied rewrites 85.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 85.4

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

9. Applied rewrites 85.4%

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

    1. lift-fma.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. add-to-fraction N/A

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

    4. mult-flip N/A

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

11. Applied rewrites 84.9%

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

12. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower--.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-PI.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. lower-PI.f32 84.9

       \[\leadsto \log \left(\frac{1}{\left(1 + 2 \cdot \frac{u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}{s}\right) \cdot \frac{1}{\frac{\pi}{s} - -2}} - 1\right) \cdot \left(-s\right) \]

14. Applied rewrites 84.9%

    \[\leadsto \log \left(\frac{1}{\color{blue}{\left(1 + 2 \cdot \frac{u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}{s}\right)} \cdot \frac{1}{\frac{\pi}{s} - -2}} - 1\right) \cdot \left(-s\right) \]
15. Add Preprocessing

Alternative 6: 25.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.8

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

4. Applied rewrites 94.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 85.4

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

7. Applied rewrites 85.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 85.4

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

9. Applied rewrites 85.4%

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

    1. lift-fma.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. add-to-fraction N/A

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

    4. mult-flip N/A

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

11. Applied rewrites 84.9%

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

12. Taylor expanded in u around 0

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

14. Applied rewrites 25.2%

    \[\leadsto \log \left(\frac{1}{\color{blue}{1} \cdot \frac{1}{\frac{\pi}{s} - -2}} - 1\right) \cdot \left(-s\right) \]
15. Add Preprocessing

Alternative 7: 14.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \pi\right) \cdot u\\ \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\left(-s\right) \cdot \log 1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(1 - \frac{0.25 \cdot \pi}{t\_0}\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* (* 0.5 PI) u)))
   (if (<= s 9.999999682655225e-21)
     (* (- s) (log 1.0))
     (* 4.0 (* (- 1.0 (/ (* 0.25 PI) t_0)) t_0)))))

C

float code(float u, float s) {
	float t_0 = (0.5f * ((float) M_PI)) * u;
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = -s * logf(1.0f);
	} else {
		tmp = 4.0f * ((1.0f - ((0.25f * ((float) M_PI)) / t_0)) * t_0);
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(Float32(0.5) * Float32(pi)) * u)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = Float32(Float32(-s) * log(Float32(1.0)));
	else
		tmp = Float32(Float32(4.0) * Float32(Float32(Float32(1.0) - Float32(Float32(Float32(0.25) * Float32(pi)) / t_0)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = (single(0.5) * single(pi)) * u;
	tmp = single(0.0);
	if (s <= single(9.999999682655225e-21))
		tmp = -s * log(single(1.0));
	else
		tmp = single(4.0) * ((single(1.0) - ((single(0.25) * single(pi)) / t_0)) * t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 10.3%

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

   if 9.99999968e-21 < s

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

      \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 4 \cdot \color{blue}{\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower--.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      6. lower-PI.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      8. lower-PI.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      10. lower-PI.f32 11.7

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

   4. Applied rewrites 11.7%

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

      1. lift--.f32 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f32 N/A

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

   6. Applied rewrites 11.7%

      \[\leadsto 4 \cdot \left(\left(1 - \frac{0.25 \cdot \pi}{\left(0.5 \cdot \pi\right) \cdot u}\right) \cdot \color{blue}{\left(\left(0.5 \cdot \pi\right) \cdot u\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 14.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\left(-s\right) \cdot \log 1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(u \cdot \left(\mathsf{fma}\left(-0.25, \frac{\pi}{u}, 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 9.999999682655225e-21)
   (* (- s) (log 1.0))
   (* 4.0 (* u (- (fma -0.25 (/ PI u) (* 0.25 PI)) (* -0.25 PI))))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = -s * logf(1.0f);
	} else {
		tmp = 4.0f * (u * (fmaf(-0.25f, (((float) M_PI) / u), (0.25f * ((float) M_PI))) - (-0.25f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = Float32(Float32(-s) * log(Float32(1.0)));
	else
		tmp = Float32(Float32(4.0) * Float32(u * Float32(fma(Float32(-0.25), Float32(Float32(pi) / u), Float32(Float32(0.25) * Float32(pi))) - Float32(Float32(-0.25) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 10.3%

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

   if 9.99999968e-21 < s

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

      \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 4 \cdot \color{blue}{\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower--.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      6. lower-PI.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      8. lower-PI.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      10. lower-PI.f32 11.7

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

   4. Applied rewrites 11.7%

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

   5. Taylor expanded in u around inf

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

      1. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{u} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      2. lower--.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{u} + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\mathsf{fma}\left(\frac{-1}{4}, \frac{\mathsf{PI}\left(\right)}{u}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      4. lower-/.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\mathsf{fma}\left(\frac{-1}{4}, \frac{\mathsf{PI}\left(\right)}{u}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. lower-PI.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\mathsf{fma}\left(\frac{-1}{4}, \frac{\pi}{u}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\mathsf{fma}\left(\frac{-1}{4}, \frac{\pi}{u}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\mathsf{fma}\left(\frac{-1}{4}, \frac{\pi}{u}, \frac{1}{4} \cdot \pi\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\mathsf{fma}\left(\frac{-1}{4}, \frac{\pi}{u}, \frac{1}{4} \cdot \pi\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      9. lower-PI.f32 11.7

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

   7. Applied rewrites 11.7%

      \[\leadsto 4 \cdot \left(u \cdot \color{blue}{\left(\mathsf{fma}\left(-0.25, \frac{\pi}{u}, 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 14.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\left(-s\right) \cdot \log 1\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \left(u \cdot \mathsf{fma}\left(-2, \frac{\pi}{s}, \frac{\pi}{s \cdot u}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 9.999999682655225e-21)
   (* (- s) (log 1.0))
   (* (- s) (* u (fma -2.0 (/ PI s) (/ PI (* s u)))))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = -s * logf(1.0f);
	} else {
		tmp = -s * (u * fmaf(-2.0f, (((float) M_PI) / s), (((float) M_PI) / (s * u))));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = Float32(Float32(-s) * log(Float32(1.0)));
	else
		tmp = Float32(Float32(-s) * Float32(u * fma(Float32(-2.0), Float32(Float32(pi) / s), Float32(Float32(pi) / Float32(s * u)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 10.3%

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

   if 9.99999968e-21 < s

   1. Initial program 98.9%

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

   2. Taylor expanded in s around -inf

      \[\leadsto \left(-s\right) \cdot \color{blue}{\left(4 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(-s\right) \cdot \left(4 \cdot \color{blue}{\frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}}\right) \]

      2. lower-/.f32 N/A

         \[\leadsto \left(-s\right) \cdot \left(4 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{\color{blue}{s}}\right) \]

   4. Applied rewrites 11.7%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. div-sub N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lift--.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. distribute-rgt-out-- N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. metadata-eval N/A

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

   6. Applied rewrites 11.7%

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

   7. Taylor expanded in u around inf

      \[\leadsto \left(-s\right) \cdot \left(u \cdot \color{blue}{\left(-2 \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{\mathsf{PI}\left(\right)}{s \cdot u}\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

         \[\leadsto \left(-s\right) \cdot \left(u \cdot \mathsf{fma}\left(-2, \frac{\mathsf{PI}\left(\right)}{\color{blue}{s}}, \frac{\mathsf{PI}\left(\right)}{s \cdot u}\right)\right) \]

      3. lower-/.f32 N/A

         \[\leadsto \left(-s\right) \cdot \left(u \cdot \mathsf{fma}\left(-2, \frac{\mathsf{PI}\left(\right)}{s}, \frac{\mathsf{PI}\left(\right)}{s \cdot u}\right)\right) \]

      4. lower-PI.f32 N/A

         \[\leadsto \left(-s\right) \cdot \left(u \cdot \mathsf{fma}\left(-2, \frac{\pi}{s}, \frac{\mathsf{PI}\left(\right)}{s \cdot u}\right)\right) \]

      5. lower-/.f32 N/A

         \[\leadsto \left(-s\right) \cdot \left(u \cdot \mathsf{fma}\left(-2, \frac{\pi}{s}, \frac{\mathsf{PI}\left(\right)}{s \cdot u}\right)\right) \]

      6. lower-PI.f32 N/A

         \[\leadsto \left(-s\right) \cdot \left(u \cdot \mathsf{fma}\left(-2, \frac{\pi}{s}, \frac{\pi}{s \cdot u}\right)\right) \]

      7. lower-*.f32 11.3

         \[\leadsto \left(-s\right) \cdot \left(u \cdot \mathsf{fma}\left(-2, \frac{\pi}{s}, \frac{\pi}{s \cdot u}\right)\right) \]

   9. Applied rewrites 11.3%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{\pi}{s}, \frac{\pi}{s \cdot u}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 14.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\left(-s\right) \cdot \log 1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \mathsf{fma}\left(u \cdot \pi, 0.5, -0.25 \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 9.999999682655225e-21)
   (* (- s) (log 1.0))
   (* 4.0 (fma (* u PI) 0.5 (* -0.25 PI)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = -s * logf(1.0f);
	} else {
		tmp = 4.0f * fmaf((u * ((float) M_PI)), 0.5f, (-0.25f * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = Float32(Float32(-s) * log(Float32(1.0)));
	else
		tmp = Float32(Float32(4.0) * fma(Float32(u * Float32(pi)), Float32(0.5), Float32(Float32(-0.25) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 10.3%

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

   if 9.99999968e-21 < s

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

      \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 4 \cdot \color{blue}{\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower--.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      6. lower-PI.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      8. lower-PI.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      10. lower-PI.f32 11.7

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

   4. Applied rewrites 11.7%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f32 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. metadata-eval 11.7

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

   6. Applied rewrites 11.7%

      \[\leadsto 4 \cdot \mathsf{fma}\left(u \cdot \pi, \color{blue}{0.5}, -0.25 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 14.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\left(-s\right) \cdot \frac{1}{\frac{s}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
           1.0)))
        -1.999999936531045e-19)
     (* (- s) (/ 1.0 (/ s PI)))
     (* (- s) (log 1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float tmp;
	if ((-s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f))) <= -1.999999936531045e-19f) {
		tmp = -s * (1.0f / (s / ((float) M_PI)));
	} else {
		tmp = -s * logf(1.0f);
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	tmp = Float32(0.0)
	if (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)))) <= Float32(-1.999999936531045e-19))
		tmp = Float32(Float32(-s) * Float32(Float32(1.0) / Float32(s / Float32(pi))));
	else
		tmp = Float32(Float32(-s) * log(Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = single(0.0);
	if ((-s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)))) <= single(-1.999999936531045e-19))
		tmp = -s * (single(1.0) / (s / single(pi)));
	else
		tmp = -s * log(single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.99999994e-19

   1. Initial program 98.9%

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

      1. lift-*.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft-in N/A

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

      5. flip3-+ N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(u \cdot \frac{1}{1 + e^{\frac{-\pi}{s}}}\right)}^{3} + {\left(u \cdot \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)\right)}^{3}}{\left(u \cdot \frac{1}{1 + e^{\frac{-\pi}{s}}}\right) \cdot \left(u \cdot \frac{1}{1 + e^{\frac{-\pi}{s}}}\right) + \left(\left(u \cdot \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)\right) \cdot \left(u \cdot \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)\right) - \left(u \cdot \frac{1}{1 + e^{\frac{-\pi}{s}}}\right) \cdot \left(u \cdot \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)\right)\right)}} + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(\frac{u}{e^{\frac{-\pi}{s}} - -1}\right)}^{3} + {\left(\frac{u}{-1 - e^{\frac{\pi}{s}}}\right)}^{3}}{\mathsf{fma}\left(\frac{u}{e^{\frac{-\pi}{s}} - -1}, \frac{u}{e^{\frac{-\pi}{s}} - -1}, \frac{u}{-1 - e^{\frac{\pi}{s}}} \cdot \frac{u}{-1 - e^{\frac{\pi}{s}}} - \frac{u}{e^{\frac{-\pi}{s}} - -1} \cdot \frac{u}{-1 - e^{\frac{\pi}{s}}}\right)}} + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

   4. Taylor expanded in u around 0

      \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
   5. Step-by-step derivation

      1. lower-/.f32 N/A

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

      2. lower-PI.f32 11.5

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

   6. Applied rewrites 11.5%

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

      1. lift-/.f32 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f32 N/A

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

      4. lower-unsound-/.f32 11.5

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

   8. Applied rewrites 11.5%

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

   if -1.99999994e-19 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 10.3%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 14.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\left(-s\right) \cdot \frac{\pi}{s}\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
           1.0)))
        -1.999999936531045e-19)
     (* (- s) (/ PI s))
     (* (- s) (log 1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float tmp;
	if ((-s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f))) <= -1.999999936531045e-19f) {
		tmp = -s * (((float) M_PI) / s);
	} else {
		tmp = -s * logf(1.0f);
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	tmp = Float32(0.0)
	if (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)))) <= Float32(-1.999999936531045e-19))
		tmp = Float32(Float32(-s) * Float32(Float32(pi) / s));
	else
		tmp = Float32(Float32(-s) * log(Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = single(0.0);
	if ((-s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)))) <= single(-1.999999936531045e-19))
		tmp = -s * (single(pi) / s);
	else
		tmp = -s * log(single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.99999994e-19

   1. Initial program 98.9%

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

   2. Taylor expanded in u around 0

      \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
   3. Step-by-step derivation

      1. lower-/.f32 N/A

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

      2. lower-PI.f32 11.5

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

   4. Applied rewrites 11.5%

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

   if -1.99999994e-19 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 10.3%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 11.5% accurate, 46.3× 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(-Float32(pi))
end

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto -1 \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

   2. lower-PI.f32 11.5

      \[\leadsto -1 \cdot \pi \]

4. Applied rewrites 11.5%

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

   1. lift-*.f32 N/A

      \[\leadsto -1 \cdot \color{blue}{\pi} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\pi\right) \]

   3. lift-neg.f32 11.5

      \[\leadsto -\pi \]

6. Applied rewrites 11.5%

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

Reproduce

herbie shell --seed 2025157 
(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))))