Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 5.0s

Alternatives: 10

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
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (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

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

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
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (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

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

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (let* ((t_0 (exp (/ PI s)))
       (t_1
        (*
         (*
          (-
           (-
            (/ 1.0 (fma t_0 u u))
            (/ -1.0 (- (exp (/ (- PI) s)) -1.0)))
           (/ -1.0 (- -1.0 t_0)))
          u)
         2.0)))
  (* (- s) (log (/ (- 2.0 t_1) t_1)))))

C

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

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(Float32(Float32(Float32(Float32(Float32(1.0) / fma(t_0, u, u)) - Float32(Float32(-1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0)))) - Float32(Float32(-1.0) / Float32(Float32(-1.0) - t_0))) * u) * Float32(2.0))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(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. Taylor expanded in u around inf

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.9%

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

8. Applied rewrites 99.0%

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

Alternative 2: 98.9% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (let* ((t_0 (exp (/ PI s))))
  (*
   (- s)
   (log
    (-
     (/
      1.0
      (*
       u
       (-
        (-
         (/ 1.0 (fma t_0 u u))
         (/ -1.0 (- (exp (/ -3.1415927410125732 s)) -1.0)))
        (/ 1.0 (- t_0 -1.0)))))
     1.0)))))

C

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

Julia

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

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.9%

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

7. Evaluated real constant 98.9%

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

Alternative 3: 97.6% accurate, 1.3× speedup

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

Math

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

FPCore

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

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

4. Applied rewrites 97.6%

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

Alternative 4: 85.6% accurate, 1.3× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{1}{2 + \frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-3.1415927410125732}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (let* ((t_0 (/ 1.0 (+ 2.0 (/ PI s)))))
  (*
   (- s)
   (log
    (-
     (/
      1.0
      (+
       (* u (- (/ 1.0 (+ 1.0 (exp (/ -3.1415927410125732 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((-3.1415927410125732f / 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(-3.1415927410125732) / 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(-3.1415927410125732) / 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

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

4. Applied rewrites 85.6%

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

5. Evaluated real constant 85.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-3.1415927410125732}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{2 + \frac{\pi}{s}}} - 1\right) \]
6. Add Preprocessing

Alternative 5: 85.0% accurate, 1.9× speedup

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

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (log
  (fma
   (/ 1.0 (+ 1.0 (* 2.0 (/ (* u (- (* 0.25 PI) (* -0.25 PI))) s))))
   (- (/ PI s) -2.0)
   -1.0))))

C

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

Julia

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

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

4. Applied rewrites 85.6%

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

6. Applied rewrites 85.0%

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

7. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{1 + 2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s}}, \frac{\pi}{s} - -2, -1\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 85.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{1 + 2 \cdot \frac{u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}{s}}, \frac{\pi}{s} - -2, -1\right)\right) \]
10. Add Preprocessing

Alternative 6: 36.0% accurate, 2.0× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{1}{2 + \frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - t\_0\right) + t\_0} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (let* ((t_0 (/ 1.0 (+ 2.0 (/ PI s)))))
  (* (- s) (log (- (/ 1.0 (+ (* u (- 0.5 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 * (0.5f - 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(0.5) - 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(0.5) - 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

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

4. Applied rewrites 85.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 36.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{2 + \frac{\pi}{s}}} - 1\right) \]
8. Add Preprocessing

Alternative 7: 25.1% accurate, 4.5× speedup

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

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (+ 2.0 (/ PI s)) 1.0))))

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(2.0) + (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

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in s around -inf

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

9. Applied rewrites 24.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + 4 \cdot \frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) - -0.25 \cdot \pi}{s}\right) - 1\right) \]

10. Taylor expanded in u around 0

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

12. Applied rewrites 25.1%

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

Alternative 8: 11.4% accurate, 5.0× speedup

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

Math

\[{-1.4645918607711792}^{3} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (pow -1.4645918607711792 3.0))

C

float code(float u, float s) {
	return powf(-1.4645918607711792f, 3.0f);
}

Fortran

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = (-1.4645918607711792e0) ** 3.0e0
end function

Julia

function code(u, s)
	return Float32(-1.4645918607711792) ^ Float32(3.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(-1.4645918607711792) ^ single(3.0);
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 0

   \[\leadsto -1 \cdot \pi \]
3. Step-by-step derivation

4. Applied rewrites 11.4%

   \[\leadsto -1 \cdot \pi \]
5. Step-by-step derivation

6. Applied rewrites 11.4%

   \[\leadsto {\left(\sqrt[3]{-\pi}\right)}^{3} \]

7. Evaluated real constant 11.4%

   \[\leadsto {-1.4645918607711792}^{3} \]
8. Add Preprocessing

Alternative 9: 11.4% accurate, 90.2× speedup

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

Math

\[-3.1415927410125732 \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  -3.1415927410125732)

C

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

Fortran

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

   \[\leadsto -1 \cdot \pi \]
3. Step-by-step derivation

4. Applied rewrites 11.4%

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

5. Evaluated real constant 11.4%

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

Reproduce

herbie shell --seed 2025359 
(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))))