Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 7.7s

Alternatives: 10

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}} - -1\\ t_1 := e^{\frac{-\pi}{s}}\\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(-\left(\log \left(1 + \frac{1}{\left(\frac{t\_0}{t\_1 - -1} - 1\right) \cdot u}\right) + \log \left(u \cdot \left(\frac{-1}{-1 - t\_1} - \frac{1}{t\_0}\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (exp (/ PI s)) -1.0)) (t_1 (exp (/ (- PI) s))))
   (*
    (- s)
    (log
     (expm1
      (-
       (+
        (log (+ 1.0 (/ 1.0 (* (- (/ t_0 (- t_1 -1.0)) 1.0) u))))
        (log (* u (- (/ -1.0 (- -1.0 t_1)) (/ 1.0 t_0)))))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s)) - -1.0f;
	float t_1 = expf((-((float) M_PI) / s));
	return -s * logf(expm1f(-(logf((1.0f + (1.0f / (((t_0 / (t_1 - -1.0f)) - 1.0f) * u)))) + logf((u * ((-1.0f / (-1.0f - t_1)) - (1.0f / t_0)))))));
}

Julia

function code(u, s)
	t_0 = Float32(exp(Float32(Float32(pi) / s)) - Float32(-1.0))
	t_1 = exp(Float32(Float32(-Float32(pi)) / s))
	return Float32(Float32(-s) * log(expm1(Float32(-Float32(log(Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(Float32(t_0 / Float32(t_1 - Float32(-1.0))) - Float32(1.0)) * u)))) + log(Float32(u * Float32(Float32(Float32(-1.0) / Float32(Float32(-1.0) - t_1)) - Float32(Float32(1.0) / t_0)))))))))
end

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 4.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{expm1}\left(-\log \left(\frac{\mathsf{fma}\left(u, \frac{e^{\frac{\pi}{s}} - -1}{e^{\frac{-\pi}{s}} - -1} - 1, 1\right)}{e^{\frac{\pi}{s}} - -1}\right)\right)\right)} \]
4. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. sum-to-mult N/A

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

   5. associate-/l* N/A

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

5. Applied rewrites 98.8%

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

Alternative 2: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - t\_0, t\_0\right)\right) \cdot -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (expm1
      (*
       (log (fma u (- (/ 1.0 (+ 1.0 (exp (* -1.0 (/ 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(expm1f((logf(fmaf(u, ((1.0f / (1.0f + expf((-1.0f * (((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(expm1(Float32(log(fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))))) - t_0), t_0)) * Float32(-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

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 3.7

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

4. Applied rewrites 3.7%

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

6. Applied rewrites 3.7%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{0.5 \cdot \pi}{s}, u, \frac{1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot -1\right)\right)} \]

7. Taylor expanded in u around 0

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \color{blue}{\left(u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} \cdot -1\right)\right) \]
8. Step-by-step derivation

   1. lower-fma.f32 N/A

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

9. Applied rewrites 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)} \cdot -1\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((((single(-1.0) / (single(-1.0) - exp((-single(pi) / s)))) - (single(1.0) / (exp((single(pi) / s)) - single(-1.0)))) * u) + (single(1.0) / (single(1.0) + exp((single(1.0) / (s / single(pi)))))))) - 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. 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. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 98.9

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 98.9

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

Alternative 5: 97.5% accurate, 1.3× speedup

Math

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

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

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.5%

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

Alternative 6: 85.9% accurate, 1.3× speedup

Math

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

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

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.8

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

4. Applied rewrites 94.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 85.9

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

7. Applied rewrites 85.9%

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

Alternative 7: 85.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	t_0 = Float32(Float32(2.0) + Float32(Float32(pi) / s))
	return Float32(Float32(-s) * log(fma(Float32(Float32(1.0) / fma(Float32(Float32(t_0 / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0))) - Float32(1.0)), u, Float32(1.0))), t_0, Float32(-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. 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. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 98.9

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 98.9

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 4.0%

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

8. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 5.8

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

10. Applied rewrites 5.8%

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

11. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-/.f32 N/A

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

    3. lower-PI.f32 85.3

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

13. Applied rewrites 85.3%

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

Alternative 8: 24.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-+.f32 N/A

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

   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. Add Preprocessing

Alternative 9: 14.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.3%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.4

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

7. Applied rewrites 14.4%

   \[\leadsto \left(-s\right) \cdot \frac{s}{\color{blue}{u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}} \]
8. Add Preprocessing

Alternative 10: 11.4% accurate, 46.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 11.4

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

4. Applied rewrites 11.4%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
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 11.4

      \[\leadsto -\pi \]

6. Applied rewrites 11.4%

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

Reproduce

herbie shell --seed 2025151 
(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))))