Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 11.6s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 99.0%

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

Alternative 2: 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

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: 97.7% 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 97.7%

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

Alternative 4: 86.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 2.0 (/ PI s)))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ t_0 (* (- (/ 1.0 (- (exp (/ (- PI) s)) -1.0)) t_0) u)))
      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 / (t_0 + (((1.0f / (expf((-((float) M_PI) / s)) - -1.0f)) - t_0) * u))) - 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(t_0 + Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0))) - t_0) * u))) - 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) / (t_0 + (((single(1.0) / (exp((-single(pi) / s)) - single(-1.0))) - t_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 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 95.0

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

4. Applied rewrites 95.0%

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

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 86.4

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

7. Applied rewrites 86.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 86.4

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 86.4

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

9. Applied rewrites 86.4%

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

Alternative 5: 86.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.9

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

8. Applied rewrites 94.9%

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

9. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-/.f32 N/A

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

    3. lower-PI.f32 83.2

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

11. Applied rewrites 83.2%

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

12. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-/.f32 N/A

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

    3. lower-PI.f32 83.3

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

14. Applied rewrites 83.3%

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

15. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-/.f32 N/A

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

    3. lower-PI.f32 86.3

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

17. Applied rewrites 86.3%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

19. Applied rewrites 86.4%

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

Alternative 6: 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 7: 14.4% 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 17.3%

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

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 14.4

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

7. Applied rewrites 14.4%

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

Alternative 8: 11.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(\left(1 - \frac{0.5}{u}\right) \cdot \left(\left(\pi \cdot 0.5\right) \cdot u\right)\right) \end{array} \]

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(1.0) - Float32(Float32(0.5) / u)) * Float32(Float32(Float32(pi) * Float32(0.5)) * u)))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(1.0) - (single(0.5) / u)) * ((single(pi) * single(0.5)) * u));
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.5

       \[\leadsto 4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right) \]

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-to-mult N/A

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

   3. lower-*.f32 N/A

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

6. Applied rewrites 11.5%

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

7. Taylor expanded in u around 0

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

   1. lower-/.f32 11.5

      \[\leadsto 4 \cdot \left(\left(1 - \frac{0.5}{u}\right) \cdot \left(\left(\pi \cdot 0.5\right) \cdot u\right)\right) \]

9. Applied rewrites 11.5%

   \[\leadsto 4 \cdot \left(\left(1 - \frac{0.5}{u}\right) \cdot \left(\left(\pi \cdot \color{blue}{0.5}\right) \cdot u\right)\right) \]
10. Add Preprocessing

Alternative 9: 11.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(\frac{u - 0.5}{u} \cdot \left(\left(\pi \cdot 0.5\right) \cdot u\right)\right) \end{array} \]

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(u - Float32(0.5)) / u) * Float32(Float32(Float32(pi) * Float32(0.5)) * u)))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (((u - single(0.5)) / u) * ((single(pi) * single(0.5)) * u));
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.5

       \[\leadsto 4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right) \]

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-to-mult N/A

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

   3. lower-*.f32 N/A

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

6. Applied rewrites 11.5%

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

7. Taylor expanded in u around 0

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

   1. lower-/.f32 N/A

      \[\leadsto 4 \cdot \left(\frac{u - \frac{1}{2}}{u} \cdot \left(\left(\pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot u\right)\right) \]

   2. lower--.f32 11.5

      \[\leadsto 4 \cdot \left(\frac{u - 0.5}{u} \cdot \left(\left(\pi \cdot 0.5\right) \cdot u\right)\right) \]

9. Applied rewrites 11.5%

   \[\leadsto 4 \cdot \left(\frac{u - 0.5}{u} \cdot \left(\color{blue}{\left(\pi \cdot 0.5\right)} \cdot u\right)\right) \]
10. Add Preprocessing

Alternative 10: 11.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ u \cdot \mathsf{fma}\left(-1, \frac{\pi}{u}, 2 \cdot \pi\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.5

       \[\leadsto 4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right) \]

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-to-mult N/A

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

   3. lower-*.f32 N/A

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

6. Applied rewrites 11.5%

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

7. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 11.5

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

9. Applied rewrites 11.5%

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

Alternative 11: 11.5% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) * ((single(0.5) * u) - single(0.25))) * single(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. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.5

       \[\leadsto 4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right) \]

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-to-mult N/A

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

   3. lower-*.f32 N/A

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

6. Applied rewrites 11.5%

   \[\leadsto 4 \cdot \left(\left(1 - \frac{0.25 \cdot \pi}{\left(\pi \cdot 0.5\right) \cdot u}\right) \cdot \color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot u\right)}\right) \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\left(1 - \frac{\frac{1}{4} \cdot \pi}{\left(\pi \cdot \frac{1}{2}\right) \cdot u}\right) \cdot \left(\left(\pi \cdot \frac{1}{2}\right) \cdot u\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(1 - \frac{\frac{1}{4} \cdot \pi}{\left(\pi \cdot \frac{1}{2}\right) \cdot u}\right) \cdot \left(\left(\pi \cdot \frac{1}{2}\right) \cdot u\right)\right) \cdot \color{blue}{4} \]

   3. lift-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. sub-to-mult-rev N/A

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

   7. *-lft-identity N/A

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

   8. lower-*.f32 N/A

      \[\leadsto \left(1 \cdot \left(\left(\pi \cdot \frac{1}{2}\right) \cdot u\right) - \frac{1}{4} \cdot \pi\right) \cdot \color{blue}{4} \]

8. Applied rewrites 11.5%

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

Alternative 12: 11.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, \pi, 2 \cdot \left(u \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (fma -1.0 PI (* 2.0 (* u PI))))

C

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

Julia

function code(u, s)
	return fma(Float32(-1.0), Float32(pi), Float32(Float32(2.0) * Float32(u * Float32(pi))))
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.5

       \[\leadsto 4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right) \]

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-to-mult N/A

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

   3. lower-*.f32 N/A

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

6. Applied rewrites 11.5%

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

7. Taylor expanded in u around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 11.5

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

9. Applied rewrites 11.5%

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

Alternative 13: 11.3% 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 11.3

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

4. Applied rewrites 11.3%

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

   1. lift-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f32 11.3

      \[\leadsto -\pi \]

6. Applied rewrites 11.3%

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

Reproduce

herbie shell --seed 2025156 
(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))))