Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 8.0s

Alternatives: 12

Speedup: 0.2×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	t_0 = Float32(exp(Float32(Float32(pi) / s)) - Float32(-1.0))
	t_1 = Float32(u * Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0))) - Float32(Float32(1.0) / t_0)))
	t_2 = Float32((t_0 ^ Float32(-3.0)) + (t_1 ^ Float32(3.0)))
	return Float32(Float32(-s) * log(Float32(fma(Float32((t_0 ^ Float32(-2.0)) + (t_1 ^ Float32(2.0))), Float32(Float32(1.0) / t_2), Float32(-Float32(t_1 / Float32(t_0 * t_2)))) - 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 98.8%

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

   1. lift--.f32 N/A

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

5. Applied rewrites 98.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

Alternative 3: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.3%

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

Alternative 4: 86.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(2.0) + (single(pi) / s));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 95.1

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

4. Applied rewrites 95.1%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 86.2

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

7. Applied rewrites 86.2%

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

Alternative 5: 85.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift--.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. sub-to-fraction N/A

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

   7. associate-*l/ N/A

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

3. Applied rewrites 4.1%

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

4. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 6.0

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

6. Applied rewrites 6.0%

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

7. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 85.6

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

9. Applied rewrites 85.6%

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

Alternative 6: 24.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log (- (+ 1.0 (/ PI s)) (* 2.0 (/ (* u (- (* 0.5 PI) (* -0.5 PI))) s))))))

C

float code(float u, float s) {
	return -s * logf(((1.0f + (((float) M_PI) / s)) - (2.0f * ((u * ((0.5f * ((float) M_PI)) - (-0.5f * ((float) M_PI)))) / s))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) + Float32(Float32(pi) / s)) - Float32(Float32(2.0) * Float32(Float32(u * Float32(Float32(Float32(0.5) * Float32(pi)) - Float32(Float32(-0.5) * Float32(pi)))) / s)))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) + (single(pi) / s)) - (single(2.0) * ((u * ((single(0.5) * single(pi)) - (single(-0.5) * 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift--.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. sub-to-fraction N/A

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

   7. associate-*l/ N/A

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

3. Applied rewrites 4.1%

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

4. Taylor expanded in s around inf

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

   1. lower--.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

6. Applied rewrites 24.8%

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

Alternative 7: 24.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around -inf

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

4. Applied rewrites 24.8%

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

Alternative 8: 20.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := \frac{1}{1 + t\_0}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_1\right) + t\_1} - 1\right) \leq -1.8100000136464297 \cdot 10^{-18}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{\frac{\mathsf{fma}\left(\pi \cdot \frac{\pi}{s}, 0.5, \pi\right)}{s} - -2}{1} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \frac{1}{u \cdot \left(0.5 - \frac{1}{t\_0 - -1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s))) (t_1 (/ 1.0 (+ 1.0 t_0))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_1)) t_1))
           1.0)))
        -1.8100000136464297e-18)
     (*
      (- s)
      (log (- (/ (- (/ (fma (* PI (/ PI s)) 0.5 PI) s) -2.0) 1.0) 1.0)))
     (* (- s) (/ 1.0 (* u (- 0.5 (/ 1.0 (- t_0 -1.0)))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = 1.0f / (1.0f + t_0);
	float tmp;
	if ((-s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_1)) + t_1)) - 1.0f))) <= -1.8100000136464297e-18f) {
		tmp = -s * logf(((((fmaf((((float) M_PI) * (((float) M_PI) / s)), 0.5f, ((float) M_PI)) / s) - -2.0f) / 1.0f) - 1.0f));
	} else {
		tmp = -s * (1.0f / (u * (0.5f - (1.0f / (t_0 - -1.0f)))));
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(Float32(1.0) / Float32(Float32(1.0) + t_0))
	tmp = Float32(0.0)
	if (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_1)) + t_1)) - Float32(1.0)))) <= Float32(-1.8100000136464297e-18))
		tmp = Float32(Float32(-s) * log(Float32(Float32(Float32(Float32(fma(Float32(Float32(pi) * Float32(Float32(pi) / s)), Float32(0.5), Float32(pi)) / s) - Float32(-2.0)) / Float32(1.0)) - Float32(1.0))));
	else
		tmp = Float32(Float32(-s) * Float32(Float32(1.0) / Float32(u * Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(t_0 - Float32(-1.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.81000001e-18

   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

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift--.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. sub-to-fraction N/A

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

      7. associate-*l/ N/A

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

   3. Applied rewrites 4.1%

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

   4. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-PI.f32 4.1

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \pi, -0.5 \cdot \frac{{\pi}^{2}}{s}\right)}{s}}{e^{\frac{-\pi}{s}} - -1} - 1, u, 1\right)}{e^{\frac{\pi}{s}} - -1}} - 1\right) \]

   6. Applied rewrites 4.1%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{\color{blue}{2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \pi, -0.5 \cdot \frac{{\pi}^{2}}{s}\right)}{s}}}{e^{\frac{-\pi}{s}} - -1} - 1, u, 1\right)}{e^{\frac{\pi}{s}} - -1}} - 1\right) \]

   7. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-PI.f32 42.9

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \pi, -0.5 \cdot \frac{{\pi}^{2}}{s}\right)}{s}}{e^{\frac{-\pi}{s}} - -1} - 1, u, 1\right)}{2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \pi, -0.5 \cdot \frac{{\pi}^{2}}{s}\right)}{s}}} - 1\right) \]

   9. Applied rewrites 42.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(\frac{2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \pi, -0.5 \cdot \frac{{\pi}^{2}}{s}\right)}{s}}{e^{\frac{-\pi}{s}} - -1} - 1, u, 1\right)}{\color{blue}{2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \pi, -0.5 \cdot \frac{{\pi}^{2}}{s}\right)}{s}}}} - 1\right) \]

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 14.4%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{1}}{2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \pi, -0.5 \cdot \frac{{\pi}^{2}}{s}\right)}{s}}} - 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 14.4%

       \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(\pi \cdot \frac{\pi}{s}, 0.5, \pi\right)}{s} - -2}{1} - 1\right)} \]

   if -1.81000001e-18 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.9%

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

   2. Taylor expanded in u around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

   4. Applied rewrites 17.2%

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

   6. Applied rewrites 17.2%

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

   7. Taylor expanded in s around inf

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

   9. Applied rewrites 16.7%

      \[\leadsto \left(-s\right) \cdot \frac{1}{u \cdot \left(0.5 - \frac{\color{blue}{1}}{e^{\frac{\pi}{s}} - -1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 16.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * (single(1.0) / (u * (single(0.5) - (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

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.2%

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

6. Applied rewrites 17.2%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 16.7%

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

Alternative 10: 16.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right)} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (/ 1.0 (* u (- 0.5 (/ 1.0 (+ 2.0 (/ PI s))))))))

C

float code(float u, float s) {
	return -s * (1.0f / (u * (0.5f - (1.0f / (2.0f + (((float) M_PI) / s))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * Float32(Float32(1.0) / Float32(u * Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * (single(1.0) / (u * (single(0.5) - (single(1.0) / (single(2.0) + (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 inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.2%

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

6. Applied rewrites 17.2%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 16.7%

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

10. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-/.f32 N/A

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

    3. lower-PI.f32 16.6

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

12. Applied rewrites 16.6%

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

Alternative 11: 14.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.3

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

7. Applied rewrites 14.3%

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

Alternative 12: 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. lower-*.f32 N/A

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

   2. lower-PI.f32 11.5

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

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f32 11.5

      \[\leadsto -\pi \]

6. Applied rewrites 11.5%

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

Reproduce

herbie shell --seed 2025140 
(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))))