Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 4.4s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

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

Alternative 2: 97.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in u around inf

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

6. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 97.6

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

8. Applied rewrites 97.6%

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

Alternative 3: 94.3% accurate, 1.6× speedup

Math

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

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 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 98.9%

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

4. Taylor expanded in u around inf

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\pi}{s}} - -1} - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \cdot u} - 1\right) \]
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. lift-/.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. lift-PI.f32 94.3

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

9. Applied rewrites 94.3%

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

Alternative 4: 37.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in u around inf

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

6. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 97.6

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

8. Applied rewrites 97.6%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 37.1%

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

Alternative 5: 25.1% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 24.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot -0.5, u, 0.25 \cdot \pi\right)}{s}, 4, 1\right)\right)} \]

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. lift-PI.f32 25.1

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

11. Applied rewrites 25.1%

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

Alternative 6: 11.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-\pi}{u}, u, \left(\pi + \pi\right) \cdot u\right) \end{array} \]

FPCore

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

C

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

Julia

function code(u, s)
	return fma(Float32(Float32(-Float32(pi)) / u), u, Float32(Float32(Float32(pi) + Float32(pi)) * u))
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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 11.5%

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

5. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-PI.f32 11.5

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

7. Applied rewrites 11.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-neg.f32 N/A

      \[\leadsto \left(2 \cdot \mathsf{PI}\left(\right) + \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{u}\right) \cdot u \]

   7. distribute-frac-neg N/A

      \[\leadsto \left(2 \cdot \mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{u}\right)\right)\right) \cdot u \]

   8. mul-1-neg N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-in N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. distribute-frac-neg N/A

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

   15. lift-neg.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. lower-*.f32 N/A

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

9. Applied rewrites 11.5%

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

Alternative 7: 11.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot \pi\right) \cdot 2 - \pi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = ((u * single(pi)) * single(2.0)) - 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 s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 11.5%

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

5. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-PI.f32 11.5

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

7. Applied rewrites 11.5%

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

8. Taylor expanded in u around inf

   \[\leadsto \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u \]
9. Step-by-step derivation

   1. count-2-rev N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u \]

   2. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u \]

   3. lift-PI.f32 N/A

      \[\leadsto \left(\pi + \mathsf{PI}\left(\right)\right) \cdot u \]

   4. lift-PI.f32 5.0

      \[\leadsto \left(\pi + \pi\right) \cdot u \]

10. Applied rewrites 5.0%

    \[\leadsto \left(\pi + \pi\right) \cdot u \]

11. Taylor expanded in u around 0

    \[\leadsto -1 \cdot \mathsf{PI}\left(\right) + \color{blue}{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

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

    2. mul-1-neg N/A

       \[\leadsto 2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \]

    3. sub-flip N/A

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

    4. lower--.f32 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lift-PI.f32 N/A

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

    9. lift-PI.f32 11.5

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

13. Applied rewrites 11.5%

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

Alternative 8: 11.3% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * (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 u around 0

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 11.2

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

4. Applied rewrites 11.2%

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

Alternative 9: 11.2% 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. mul-1-neg N/A

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

   2. lift-neg.f32 N/A

      \[\leadsto -\mathsf{PI}\left(\right) \]

   3. lift-PI.f32 11.3

      \[\leadsto -\pi \]

4. Applied rewrites 11.3%

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

Reproduce

herbie shell --seed 2025129 
(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))))