Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 12.2s

Alternatives: 10

Speedup: 0.8×

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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ t_1 := 1 - e^{-2 \cdot \frac{\pi}{s}}\\ t_2 := \mathsf{fma}\left(u, \frac{1}{t\_1} - \left(t\_0 + \frac{e^{-1 \cdot \frac{\pi}{s}}}{t\_1}\right), t\_0\right)\\ \left(-s\right) \cdot \log \left(\frac{{t\_2}^{-3} - 1}{1 + \frac{\mathsf{fma}\left(1, {t\_2}^{2}, t\_2 \cdot 1\right)}{{t\_2}^{3}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s)))))
        (t_1 (- 1.0 (exp (* -2.0 (/ PI s)))))
        (t_2
         (fma u (- (/ 1.0 t_1) (+ t_0 (/ (exp (* -1.0 (/ PI s))) t_1))) t_0)))
   (*
    (- s)
    (log
     (/
      (- (pow t_2 -3.0) 1.0)
      (+ 1.0 (/ (fma 1.0 (pow t_2 2.0) (* t_2 1.0)) (pow t_2 3.0))))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float t_1 = 1.0f - expf((-2.0f * (((float) M_PI) / s)));
	float t_2 = fmaf(u, ((1.0f / t_1) - (t_0 + (expf((-1.0f * (((float) M_PI) / s))) / t_1))), t_0);
	return -s * logf(((powf(t_2, -3.0f) - 1.0f) / (1.0f + (fmaf(1.0f, powf(t_2, 2.0f), (t_2 * 1.0f)) / powf(t_2, 3.0f)))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	t_1 = Float32(Float32(1.0) - exp(Float32(Float32(-2.0) * Float32(Float32(pi) / s))))
	t_2 = fma(u, Float32(Float32(Float32(1.0) / t_1) - Float32(t_0 + Float32(exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))) / t_1))), t_0)
	return Float32(Float32(-s) * log(Float32(Float32((t_2 ^ Float32(-3.0)) - Float32(1.0)) / Float32(Float32(1.0) + Float32(fma(Float32(1.0), (t_2 ^ Float32(2.0)), Float32(t_2 * Float32(1.0))) / (t_2 ^ Float32(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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. flip-+ N/A

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

   7. lower-/.f32 N/A

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{\frac{1 - e^{\frac{-\pi}{s} \cdot 2}}{1 - e^{\frac{-\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{\frac{1 - e^{\frac{-\pi}{s} \cdot 2}}{1 - e^{\frac{-\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}, \frac{1}{\mathsf{fma}\left(u, \frac{1}{\frac{1 - e^{\frac{-\pi}{s} \cdot 2}}{1 - e^{\frac{-\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}, 1 + \frac{1}{\mathsf{fma}\left(u, \frac{1}{\frac{1 - e^{\frac{-\pi}{s} \cdot 2}}{1 - e^{\frac{-\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} \cdot 1\right)}\right)} \]

6. Taylor expanded in s around 0

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

8. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1}{1 + \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + {\left(\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-2}\right)}\right)} \]

10. Applied rewrites 98.9%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{{\left(\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1}{1 + \frac{\mathsf{fma}\left(1, {\left(\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}, \mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right) \cdot 1\right)}{\color{blue}{{\left(\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}}}}\right) \]
11. Add Preprocessing

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ t_1 := 1 - e^{-2 \cdot \frac{\pi}{s}}\\ t_2 := \mathsf{fma}\left(u, \frac{1}{t\_1} - \left(t\_0 + \frac{e^{-1 \cdot \frac{\pi}{s}}}{t\_1}\right), t\_0\right)\\ \left(-s\right) \cdot \log \left(\frac{{t\_2}^{-3} - 1}{1 + \left(\frac{1}{t\_2} + {t\_2}^{-2}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s)))))
        (t_1 (- 1.0 (exp (* -2.0 (/ PI s)))))
        (t_2
         (fma u (- (/ 1.0 t_1) (+ t_0 (/ (exp (* -1.0 (/ PI s))) t_1))) t_0)))
   (*
    (- s)
    (log (/ (- (pow t_2 -3.0) 1.0) (+ 1.0 (+ (/ 1.0 t_2) (pow t_2 -2.0))))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float t_1 = 1.0f - expf((-2.0f * (((float) M_PI) / s)));
	float t_2 = fmaf(u, ((1.0f / t_1) - (t_0 + (expf((-1.0f * (((float) M_PI) / s))) / t_1))), t_0);
	return -s * logf(((powf(t_2, -3.0f) - 1.0f) / (1.0f + ((1.0f / t_2) + powf(t_2, -2.0f)))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	t_1 = Float32(Float32(1.0) - exp(Float32(Float32(-2.0) * Float32(Float32(pi) / s))))
	t_2 = fma(u, Float32(Float32(Float32(1.0) / t_1) - Float32(t_0 + Float32(exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))) / t_1))), t_0)
	return Float32(Float32(-s) * log(Float32(Float32((t_2 ^ Float32(-3.0)) - Float32(1.0)) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / t_2) + (t_2 ^ Float32(-2.0)))))))
end

Derivation

1. Initial program 98.9%

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

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. flip-+ N/A

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

   7. lower-/.f32 N/A

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{\frac{1 - e^{\frac{-\pi}{s} \cdot 2}}{1 - e^{\frac{-\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{\frac{1 - e^{\frac{-\pi}{s} \cdot 2}}{1 - e^{\frac{-\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}, \frac{1}{\mathsf{fma}\left(u, \frac{1}{\frac{1 - e^{\frac{-\pi}{s} \cdot 2}}{1 - e^{\frac{-\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}, 1 + \frac{1}{\mathsf{fma}\left(u, \frac{1}{\frac{1 - e^{\frac{-\pi}{s} \cdot 2}}{1 - e^{\frac{-\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} \cdot 1\right)}\right)} \]

6. Taylor expanded in s around 0

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

8. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1}{1 + \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + {\left(\mathsf{fma}\left(u, \frac{1}{1 - e^{-2 \cdot \frac{\pi}{s}}} - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{e^{-1 \cdot \frac{\pi}{s}}}{1 - e^{-2 \cdot \frac{\pi}{s}}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-2}\right)}\right)} \]
9. Add Preprocessing

Alternative 3: 98.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (fma
        u
        (-
         (/ (- 1.0 (exp (* -1.0 (/ PI s)))) (- 1.0 (exp (* -2.0 (/ 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(((1.0f / fmaf(u, (((1.0f - expf((-1.0f * (((float) M_PI) / s)))) / (1.0f - expf((-2.0f * (((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(Float32(Float32(Float32(1.0) / fma(u, Float32(Float32(Float32(Float32(1.0) - exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s)))) / Float32(Float32(1.0) - exp(Float32(Float32(-2.0) * 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. flip-+ N/A

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

   7. lower-/.f32 N/A

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift--.f32 98.9

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

8. Applied rewrites 98.9%

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

Alternative 4: 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%

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

Alternative 5: 97.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (*
      (- (/ 1.0 (+ (exp (/ (- PI) 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((-((float) M_PI) / 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(-Float32(pi)) / 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(pi) / s)) + single(1.0))) - (single(1.0) / (exp((single(pi) / s)) + single(1.0)))) * u)) - single(1.0)));
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

4. Applied rewrites 97.5%

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

Alternative 6: 24.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, u, -0.25 \cdot \pi\right)}{s}, -4, 1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (fma (/ (fma (* PI 0.5) u (* -0.25 PI)) s) -4.0 1.0))))

C

float code(float u, float s) {
	return -s * logf(fmaf((fmaf((((float) M_PI) * 0.5f), u, (-0.25f * ((float) M_PI))) / s), -4.0f, 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(fma(Float32(fma(Float32(Float32(pi) * Float32(0.5)), u, Float32(Float32(-0.25) * Float32(pi))) / s), Float32(-4.0), 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

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

   1. +-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

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

4. Applied rewrites 24.9%

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

Alternative 7: 14.5% 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

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

4. Applied rewrites 17.3%

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

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. lift-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. lift-*.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. lift-PI.f32 14.5

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

7. Applied rewrites 14.5%

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

Alternative 8: 11.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi \cdot 0.5, u, -0.25 \cdot \pi\right) \cdot 4 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (fma (* PI 0.5) u (* -0.25 PI)) 4.0))

C

float code(float u, float s) {
	return fmaf((((float) M_PI) * 0.5f), u, (-0.25f * ((float) M_PI))) * 4.0f;
}

Julia

function code(u, s)
	return Float32(fma(Float32(Float32(pi) * Float32(0.5)), u, Float32(Float32(-0.25) * Float32(pi))) * Float32(4.0))
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

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
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 11.7%

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

Alternative 9: 11.5% 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 11.5

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

4. Applied rewrites 11.5%

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

Alternative 10: 11.5% 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 11.5

      \[\leadsto -\pi \]

4. Applied rewrites 11.5%

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

Reproduce

herbie shell --seed 2025143 
(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))))