Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 5.0s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

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: 99.0% 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, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{e^{\frac{\pi}{s}} - -1}\\ t_1 := \mathsf{fma}\left(u, \frac{-1}{-1 - e^{\frac{-3.1415927410125732}{s}}} - t\_0, t\_0\right)\\ \left(-s\right) \cdot \log \left(\left(1 - t\_1\right) \cdot \frac{1}{t\_1}\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1.0 (- (exp (/ PI s)) -1.0)))
       (t_1
        (fma
         u
         (- (/ -1.0 (- -1.0 (exp (/ -3.1415927410125732 s)))) t_0)
         t_0)))
  (* (- s) (log (* (- 1.0 t_1) (/ 1.0 t_1))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (expf((((float) M_PI) / s)) - -1.0f);
	float t_1 = fmaf(u, ((-1.0f / (-1.0f - expf((-3.1415927410125732f / s)))) - t_0), t_0);
	return -s * logf(((1.0f - t_1) * (1.0f / t_1)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) - Float32(-1.0)))
	t_1 = fma(u, Float32(Float32(Float32(-1.0) / Float32(Float32(-1.0) - exp(Float32(Float32(-3.1415927410125732) / s)))) - t_0), t_0)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) - t_1) * Float32(Float32(1.0) / t_1))))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

3. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\color{blue}{\frac{1}{s} \cdot \pi}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

5. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.0%

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

7. Applied rewrites 99.0%

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

8. Evaluated real constant 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\color{blue}{-3.1415927410125732}}{s}} - -1} - \frac{1}{e^{\frac{\pi}{s}} - -1}\right) \cdot u + \frac{1}{1 + e^{\frac{1}{s} \cdot \pi}}} - 1\right) \]

10. Applied rewrites 99.0%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 - \mathsf{fma}\left(u, \frac{-1}{-1 - e^{\frac{-3.1415927410125732}{s}}} - \frac{1}{e^{\frac{\pi}{s}} - -1}, \frac{1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot \frac{1}{\mathsf{fma}\left(u, \frac{-1}{-1 - e^{\frac{-3.1415927410125732}{s}}} - \frac{1}{e^{\frac{\pi}{s}} - -1}, \frac{1}{e^{\frac{\pi}{s}} - -1}\right)}\right)} \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-3.1415927410125732}{s}} - -1} - \frac{1}{e^{\frac{\pi}{s}} - -1}\right) \cdot u + \frac{1}{1 + e^{\frac{1}{s} \cdot \pi}}} - 1\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  (*
 (- s)
 (log
  (-
   (/
    1.0
    (+
     (*
      (-
       (/ 1.0 (- (exp (/ -3.1415927410125732 s)) -1.0))
       (/ 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 / (expf((-3.1415927410125732f / s)) - -1.0f)) - (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(exp(Float32(Float32(-3.1415927410125732) / s)) - Float32(-1.0))) - 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(Float32(1.0) / s) * Float32(pi))))))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((((single(1.0) / (exp((single(-3.1415927410125732) / s)) - single(-1.0))) - (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 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

3. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\color{blue}{\frac{1}{s} \cdot \pi}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

5. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.0%

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

7. Applied rewrites 99.0%

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

8. Evaluated real constant 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\color{blue}{-3.1415927410125732}}{s}} - -1} - \frac{1}{e^{\frac{\pi}{s}} - -1}\right) \cdot u + \frac{1}{1 + e^{\frac{1}{s} \cdot \pi}}} - 1\right) \]
9. Add Preprocessing

Alternative 3: 97.6% accurate, 1.4× speedup

Math

\[\left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-3.1415927410125732}{s}} - -1} - \frac{1}{e^{\frac{\pi}{s}} - -1}\right) \cdot u} - 1\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  (*
 (- s)
 (log
  (-
   (/
    1.0
    (*
     (-
      (/ 1.0 (- (exp (/ -3.1415927410125732 s)) -1.0))
      (/ 1.0 (- (exp (/ PI s)) -1.0)))
     u))
   1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / (((1.0f / (expf((-3.1415927410125732f / s)) - -1.0f)) - (1.0f / (expf((((float) M_PI) / s)) - -1.0f))) * u)) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-3.1415927410125732) / s)) - Float32(-1.0))) - Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) - Float32(-1.0)))) * u)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / (((single(1.0) / (exp((single(-3.1415927410125732) / s)) - single(-1.0))) - (single(1.0) / (exp((single(pi) / s)) - single(-1.0)))) * u)) - single(1.0)));
end

Derivation

1. Initial program 99.0%

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

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%

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

6. Applied rewrites 97.6%

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

7. Evaluated real constant 97.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-3.1415927410125732}{s}} - -1} - \frac{1}{e^{\frac{\pi}{s}} - -1}\right) \cdot u} - 1\right) \]
8. Add Preprocessing

Alternative 4: 94.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

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%

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.5%

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

9. Applied rewrites 94.5%

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

Alternative 5: 37.1% accurate, 2.0× speedup

Math

\[\left(-s\right) \cdot \log \left(\frac{1}{\left(0.5 - \frac{1}{e^{\frac{\pi}{s}} - -1}\right) \cdot u} - 1\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  (*
 (- s)
 (log (- (/ 1.0 (* (- 0.5 (/ 1.0 (- (exp (/ PI s)) -1.0))) u)) 1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((0.5f - (1.0f / (expf((((float) M_PI) / s)) - -1.0f))) * u)) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) - Float32(-1.0)))) * u)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((single(0.5) - (single(1.0) / (exp((single(pi) / s)) - single(-1.0)))) * u)) - single(1.0)));
end

Derivation

1. Initial program 99.0%

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

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%

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 37.1%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(0.5 - \frac{1}{e^{\frac{\pi}{s}} - -1}\right) \cdot u} - 1\right) \]
10. Add Preprocessing

Alternative 6: 25.1% accurate, 5.2× 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 99.0%

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

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 24.9%

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

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 25.1%

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

7. Applied rewrites 25.1%

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

Alternative 7: 11.3% accurate, 90.2× 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 99.0%

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

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 11.3%

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

4. Applied rewrites 11.3%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
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 11.3%

      \[\leadsto -\pi \]

6. Applied rewrites 11.3%

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

7. Evaluated real constant 11.3%

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

Reproduce

herbie shell --seed 2025212 
(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))))