Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 8.7s

Alternatives: 12

Speedup: 0.6×

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

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

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{-1 - e^{\frac{-\pi}{s}}}\\ t_1 := {e}^{\left(\frac{\pi}{s}\right)} - -1\\ t_2 := \frac{1}{t\_1}\\ \left(-s\right) \cdot \left(\log \left(\left(\left(\frac{-1}{t\_1} - u \cdot \left(t\_0 - t\_2\right)\right) + 1\right) \cdot 2\right) - \log \left(\left(t\_2 - \left(t\_2 - t\_0\right) \cdot u\right) \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ -1.0 (- -1.0 (exp (/ (- PI) s)))))
       (t_1 (- (pow E (/ PI s)) -1.0))
       (t_2 (/ 1.0 t_1)))
  (*
   (- s)
   (-
    (log (* (+ (- (/ -1.0 t_1) (* u (- t_0 t_2))) 1.0) 2.0))
    (log (* (- t_2 (* (- t_2 t_0) u)) 2.0))))))

C

float code(float u, float s) {
	float t_0 = -1.0f / (-1.0f - expf((-((float) M_PI) / s)));
	float t_1 = powf(((float) M_E), (((float) M_PI) / s)) - -1.0f;
	float t_2 = 1.0f / t_1;
	return -s * (logf(((((-1.0f / t_1) - (u * (t_0 - t_2))) + 1.0f) * 2.0f)) - logf(((t_2 - ((t_2 - t_0) * u)) * 2.0f)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(-1.0) / Float32(Float32(-1.0) - exp(Float32(Float32(-Float32(pi)) / s))))
	t_1 = Float32((Float32(exp(1)) ^ Float32(Float32(pi) / s)) - Float32(-1.0))
	t_2 = Float32(Float32(1.0) / t_1)
	return Float32(Float32(-s) * Float32(log(Float32(Float32(Float32(Float32(Float32(-1.0) / t_1) - Float32(u * Float32(t_0 - t_2))) + Float32(1.0)) * Float32(2.0))) - log(Float32(Float32(t_2 - Float32(Float32(t_2 - t_0) * u)) * Float32(2.0)))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(-1.0) / (single(-1.0) - exp((-single(pi) / s)));
	t_1 = (single(2.71828182845904523536) ^ (single(pi) / s)) - single(-1.0);
	t_2 = single(1.0) / t_1;
	tmp = -s * (log(((((single(-1.0) / t_1) - (u * (t_0 - t_2))) + single(1.0)) * single(2.0))) - log(((t_2 - ((t_2 - t_0) * u)) * single(2.0))));
end

Derivation

1. Initial program 98.9%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

5. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9%

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

7. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9%

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

9. Applied rewrites 98.9%

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

    1. lift--.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. metadata-eval N/A

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

    4. frac-sub N/A

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

    5. lower-/.f32 N/A

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

11. Applied rewrites 99.0%

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

13. Applied rewrites 98.9%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {e}^{\left(\frac{\pi}{s}\right)} - -1\\ t_1 := \left(\left(\frac{1}{e^{\frac{-\pi}{s}} - -1} - \frac{1}{t\_0}\right) \cdot u - \frac{-1}{t\_0}\right) \cdot 2\\ \left(-s\right) \cdot \log \left(\frac{2 - t\_1}{t\_1}\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (- (pow E (/ PI s)) -1.0))
       (t_1
        (*
         (-
          (* (- (/ 1.0 (- (exp (/ (- PI) s)) -1.0)) (/ 1.0 t_0)) u)
          (/ -1.0 t_0))
         2.0)))
  (* (- s) (log (/ (- 2.0 t_1) t_1)))))

C

float code(float u, float s) {
	float t_0 = powf(((float) M_E), (((float) M_PI) / s)) - -1.0f;
	float t_1 = ((((1.0f / (expf((-((float) M_PI) / s)) - -1.0f)) - (1.0f / t_0)) * u) - (-1.0f / t_0)) * 2.0f;
	return -s * logf(((2.0f - t_1) / t_1));
}

Julia

function code(u, s)
	t_0 = Float32((Float32(exp(1)) ^ Float32(Float32(pi) / s)) - Float32(-1.0))
	t_1 = Float32(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0))) - Float32(Float32(1.0) / t_0)) * u) - Float32(Float32(-1.0) / t_0)) * Float32(2.0))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(2.0) - t_1) / t_1)))
end

MATLAB

function tmp = code(u, s)
	t_0 = (single(2.71828182845904523536) ^ (single(pi) / s)) - single(-1.0);
	t_1 = ((((single(1.0) / (exp((-single(pi) / s)) - single(-1.0))) - (single(1.0) / t_0)) * u) - (single(-1.0) / t_0)) * single(2.0);
	tmp = -s * log(((single(2.0) - t_1) / t_1));
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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

5. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9%

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

7. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9%

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

9. Applied rewrites 98.9%

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

    1. lift--.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. metadata-eval N/A

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

    4. frac-sub N/A

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

    5. lower-/.f32 N/A

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

11. Applied rewrites 99.0%

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

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {e}^{\left(\frac{\pi}{s}\right)} - -1\\ t_1 := \frac{-1}{t\_0}\\ t_2 := \left(\frac{1}{e^{\frac{-\pi}{s}} - -1} - \frac{1}{t\_0}\right) \cdot u\\ \left(-s\right) \cdot \log \left(\left(1 - \left(t\_2 - t\_1\right)\right) \cdot \frac{-1}{t\_1 - t\_2}\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (- (pow E (/ PI s)) -1.0))
       (t_1 (/ -1.0 t_0))
       (t_2
        (* (- (/ 1.0 (- (exp (/ (- PI) s)) -1.0)) (/ 1.0 t_0)) u)))
  (* (- s) (log (* (- 1.0 (- t_2 t_1)) (/ -1.0 (- t_1 t_2)))))))

C

float code(float u, float s) {
	float t_0 = powf(((float) M_E), (((float) M_PI) / s)) - -1.0f;
	float t_1 = -1.0f / t_0;
	float t_2 = ((1.0f / (expf((-((float) M_PI) / s)) - -1.0f)) - (1.0f / t_0)) * u;
	return -s * logf(((1.0f - (t_2 - t_1)) * (-1.0f / (t_1 - t_2))));
}

Julia

function code(u, s)
	t_0 = Float32((Float32(exp(1)) ^ Float32(Float32(pi) / s)) - Float32(-1.0))
	t_1 = Float32(Float32(-1.0) / t_0)
	t_2 = Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0))) - Float32(Float32(1.0) / t_0)) * u)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) - Float32(t_2 - t_1)) * Float32(Float32(-1.0) / Float32(t_1 - t_2)))))
end

MATLAB

function tmp = code(u, s)
	t_0 = (single(2.71828182845904523536) ^ (single(pi) / s)) - single(-1.0);
	t_1 = single(-1.0) / t_0;
	t_2 = ((single(1.0) / (exp((-single(pi) / s)) - single(-1.0))) - (single(1.0) / t_0)) * u;
	tmp = -s * log(((single(1.0) - (t_2 - t_1)) * (single(-1.0) / (t_1 - t_2))));
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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

5. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9%

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

7. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9%

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

9. Applied rewrites 98.9%

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

11. Applied rewrites 99.0%

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

Alternative 4: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * 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)));
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 inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 97.6%

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

Alternative 5: 89.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := \frac{1}{2 + \frac{\pi}{s}}\\ \mathbf{if}\;s \leq 0.029999999329447746:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_1\right) + t\_1} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{\frac{-1 - t\_0}{\left(1 - t\_0\right) \cdot u - 1}}} - 1\right)\\ \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (exp (/ PI s))) (t_1 (/ 1.0 (+ 2.0 (/ PI s)))))
  (if (<= s 0.029999999329447746)
    (*
     (- s)
     (log
      (-
       (/
        1.0
        (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_1)) t_1))
       1.0)))
    (*
     (- s)
     (log
      (-
       (/ 1.0 (/ 1.0 (/ (- -1.0 t_0) (- (* (- 1.0 t_0) u) 1.0))))
       1.0))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = 1.0f / (2.0f + (((float) M_PI) / s));
	float tmp;
	if (s <= 0.029999999329447746f) {
		tmp = -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_1)) + t_1)) - 1.0f));
	} else {
		tmp = -s * logf(((1.0f / (1.0f / ((-1.0f - t_0) / (((1.0f - t_0) * u) - 1.0f)))) - 1.0f));
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s)))
	tmp = Float32(0.0)
	if (s <= Float32(0.029999999329447746))
		tmp = 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_1)) + t_1)) - Float32(1.0))));
	else
		tmp = Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(Float32(Float32(-1.0) - t_0) / Float32(Float32(Float32(Float32(1.0) - t_0) * u) - Float32(1.0))))) - Float32(1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = single(1.0) / (single(2.0) + (single(pi) / s));
	tmp = single(0.0);
	if (s <= single(0.029999999329447746))
		tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_1)) + t_1)) - single(1.0)));
	else
		tmp = -s * log(((single(1.0) / (single(1.0) / ((single(-1.0) - t_0) / (((single(1.0) - t_0) * u) - single(1.0))))) - single(1.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.0299999993

   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

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   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 95.1%

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}} - 1\right) \]
   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 86.4%

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

   7. Applied rewrites 86.4%

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

   if 0.0299999993 < 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. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 98.9%

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

   3. Applied rewrites 98.9%

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

      1. lift-/.f32 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 98.9%

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 3.8%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f32 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. lift--.f32 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f32 N/A

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

      11. sub-div N/A

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

      12. lower-/.f32 N/A

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

   9. Applied rewrites 3.8%

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

       1. lift-/.f32 N/A

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

       2. div-flip N/A

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

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

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

       4. lower-unsound-/.f32 3.8%

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

   11. Applied rewrites 3.8%

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

Alternative 6: 89.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2 + \frac{\pi}{s}}\\ t_1 := e^{\frac{\pi}{s}}\\ \mathbf{if}\;s \leq 0.029999999329447746:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{t\_1 - -1}{1 - \left(1 - t\_1\right) \cdot u} - 1\right)\\ \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1.0 (+ 2.0 (/ PI s)))) (t_1 (exp (/ PI s))))
  (if (<= s 0.029999999329447746)
    (*
     (- s)
     (log
      (-
       (/
        1.0
        (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
       1.0)))
    (*
     (- s)
     (log (- (/ (- t_1 -1.0) (- 1.0 (* (- 1.0 t_1) u))) 1.0))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (2.0f + (((float) M_PI) / s));
	float t_1 = expf((((float) M_PI) / s));
	float tmp;
	if (s <= 0.029999999329447746f) {
		tmp = -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
	} else {
		tmp = -s * logf((((t_1 - -1.0f) / (1.0f - ((1.0f - t_1) * u))) - 1.0f));
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s)))
	t_1 = exp(Float32(Float32(pi) / s))
	tmp = Float32(0.0)
	if (s <= Float32(0.029999999329447746))
		tmp = 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))));
	else
		tmp = Float32(Float32(-s) * log(Float32(Float32(Float32(t_1 - Float32(-1.0)) / Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - t_1) * u))) - Float32(1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = single(1.0) / (single(2.0) + (single(pi) / s));
	t_1 = exp((single(pi) / s));
	tmp = single(0.0);
	if (s <= single(0.029999999329447746))
		tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
	else
		tmp = -s * log((((t_1 - single(-1.0)) / (single(1.0) - ((single(1.0) - t_1) * u))) - single(1.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.0299999993

   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

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   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 95.1%

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}} - 1\right) \]
   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 86.4%

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

   7. Applied rewrites 86.4%

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

   if 0.0299999993 < 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. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 98.9%

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

   3. Applied rewrites 98.9%

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

      1. lift-/.f32 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 98.9%

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 3.8%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f32 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. lift--.f32 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f32 N/A

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

      11. sub-div N/A

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

      12. lower-/.f32 N/A

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

   9. Applied rewrites 3.8%

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

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

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

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

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

       3. div-flip N/A

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

       4. frac-2neg N/A

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

       5. lift--.f32 N/A

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

       6. sub-negate-rev N/A

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

       7. metadata-eval N/A

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

       8. add-flip-rev N/A

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

       9. +-commutative N/A

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

       10. lift-exp.f32 N/A

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

       11. lift-/.f32 N/A

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

       12. lift-PI.f32 N/A

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

       13. lower-/.f32 N/A

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

   11. Applied rewrites 3.8%

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

Alternative 7: 86.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1.0 (+ 2.0 (/ 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 / (2.0f + (((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(2.0) + 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(2.0) + (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. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 95.1%

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

4. Applied rewrites 95.1%

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

5. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}} - 1\right) \]
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 86.4%

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

7. Applied rewrites 86.4%

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

Alternative 8: 25.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around -inf

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

4. Applied rewrites 24.8%

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

5. Taylor expanded in u around 0

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 25.0%

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

7. Applied rewrites 25.0%

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

Alternative 9: 11.5% accurate, 9.1× speedup

Math

\[\begin{array}{l} t_0 := \left(0.25 \cdot \left(\pi + \pi\right)\right) \cdot u\\ 4 \cdot \left(\left(1 - \frac{0.25 \cdot \pi}{t\_0}\right) \cdot t\_0\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (* (* 0.25 (+ PI PI)) u)))
  (* 4.0 (* (- 1.0 (/ (* 0.25 PI) t_0)) t_0))))

C

float code(float u, float s) {
	float t_0 = (0.25f * (((float) M_PI) + ((float) M_PI))) * u;
	return 4.0f * ((1.0f - ((0.25f * ((float) M_PI)) / t_0)) * t_0);
}

Julia

function code(u, s)
	t_0 = Float32(Float32(Float32(0.25) * Float32(Float32(pi) + Float32(pi))) * u)
	return Float32(Float32(4.0) * Float32(Float32(Float32(1.0) - Float32(Float32(Float32(0.25) * Float32(pi)) / t_0)) * t_0))
end

MATLAB

function tmp = code(u, s)
	t_0 = (single(0.25) * (single(pi) + single(pi))) * u;
	tmp = single(4.0) * ((single(1.0) - ((single(0.25) * single(pi)) / t_0)) * t_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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

5. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9%

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

7. Applied rewrites 98.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.9%

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

9. Applied rewrites 98.9%

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

10. Taylor expanded in s around inf

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

    1. lower-*.f32 N/A

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

    2. lower--.f32 N/A

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

12. Applied rewrites 11.5%

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

    1. lift--.f32 N/A

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

    2. sub-to-mult N/A

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

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

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

14. Applied rewrites 11.5%

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

Alternative 10: 11.5% accurate, 26.8× speedup

Math

\[-1 \cdot \pi - -2 \cdot \left(u \cdot \pi\right) \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(-1.0) * single(pi)) - (single(-2.0) * (u * 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.9%

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 3.8%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f32 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. lift--.f32 N/A

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

   9. sub-negate-rev N/A

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

   10. lift--.f32 N/A

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

   11. sub-div N/A

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

   12. lower-/.f32 N/A

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

9. Applied rewrites 3.8%

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

10. Taylor expanded in s around inf

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

    1. lower--.f32 N/A

       \[\leadsto -1 \cdot \mathsf{PI}\left(\right) - \color{blue}{-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)} \]

    2. lower-*.f32 N/A

       \[\leadsto -1 \cdot \mathsf{PI}\left(\right) - \color{blue}{-2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) \]

    3. lower-PI.f32 N/A

       \[\leadsto -1 \cdot \pi - -2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) \]

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lower-PI.f32 11.5%

       \[\leadsto -1 \cdot \pi - -2 \cdot \left(u \cdot \pi\right) \]

12. Applied rewrites 11.5%

    \[\leadsto \color{blue}{-1 \cdot \pi - -2 \cdot \left(u \cdot \pi\right)} \]
13. Add Preprocessing

Alternative 11: 11.2% accurate, 510.0× speedup

Math

\[-3.1415927410125732 \]

FPCore

(FPCore (u s)
  :precision binary32
  -3.1415927410125732)

C

float code(float u, float s) {
	return -3.1415927410125732f;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -3.1415927410125732e0
end function

Julia

function code(u, s)
	return Float32(-3.1415927410125732)
end

MATLAB

function tmp = code(u, s)
	tmp = single(-3.1415927410125732);
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

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 11.2%

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

4. Applied rewrites 11.2%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
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 11.2%

      \[\leadsto -\pi \]

6. Applied rewrites 11.2%

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

7. Evaluated real constant 11.2%

   \[\leadsto -3.1415927410125732 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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))))