Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 5.0s

Alternatives: 11

Speedup: 1.3×

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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = (u / (exp((-single(pi) / s)) - single(-1.0))) - ((u - single(1.0)) / (exp((single(pi) / s)) - single(-1.0)));
	tmp = -s * log((((t_0 ^ single(-2.0)) - (single(-1.0) * single(-1.0))) / ((single(1.0) / 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. 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. add-flip N/A

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

   3. lower--.f32 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

   4. lower-unsound-/.f32 98.9

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

5. Applied rewrites 98.9%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--l+ N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lower-fma.f32 N/A

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 98.9%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((1.0f / fmaf(-1.0f, (-1.0f / ((expf((-((float) M_PI) / s)) - -1.0f) / u)), ((u - 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) / fma(Float32(-1.0), Float32(Float32(-1.0) / Float32(Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0)) / u)), Float32(Float32(u - Float32(1.0)) / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / s)))))) - 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. add-flip N/A

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

   3. lower--.f32 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

   4. lower-unsound-/.f32 98.9

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

5. Applied rewrites 98.9%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--l+ N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lower-fma.f32 N/A

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

7. Applied rewrites 98.9%

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

Alternative 3: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((1.0f / (((1.0f - u) / (expf((((float) M_PI) / s)) - -1.0f)) - (u / (-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(Float32(Float32(Float32(1.0) - u) / Float32(exp(Float32(Float32(pi) / s)) - Float32(-1.0))) - Float32(u / Float32(Float32(-1.0) - exp(Float32(Float32(-Float32(pi)) / s)))))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / (((single(1.0) - u) / (exp((single(pi) / s)) - single(-1.0))) - (u / (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. 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. add-flip N/A

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

   3. lower--.f32 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

   4. lower-unsound-/.f32 98.9

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

5. Applied rewrites 98.9%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--l+ N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lower-fma.f32 N/A

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 98.9%

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

Alternative 4: 86.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / fma(Float32(-1.0), Float32(Float32(-1.0) / Float32(Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0)) / u)), Float32(Float32(u - Float32(1.0)) / Float32(Float32(-1.0) - Float32(Float32(1.0) + Float32(Float32(pi) / s)))))) - 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. add-flip N/A

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

   3. lower--.f32 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

   4. lower-unsound-/.f32 98.9

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

5. Applied rewrites 98.9%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--l+ N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lower-fma.f32 N/A

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 86.2

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

10. Applied rewrites 86.2%

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

Alternative 5: 37.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((1.0f / fmaf(-1.0f, (-0.5f * u), ((u - 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) / fma(Float32(-1.0), Float32(Float32(-0.5) * u), Float32(Float32(u - Float32(1.0)) / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / s)))))) - 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. add-flip N/A

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

   3. lower--.f32 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

   4. lower-unsound-/.f32 98.9

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

5. Applied rewrites 98.9%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--l+ N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lower-fma.f32 N/A

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in s around inf

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

   1. lower-*.f32 37.5

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

10. Applied rewrites 37.5%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(-1, \color{blue}{-0.5 \cdot u}, \frac{u - 1}{-1 - e^{\frac{\pi}{s}}}\right)} - 1\right) \]
11. 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: 16.7% 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. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 17.2

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

7. Applied rewrites 17.2%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 16.7%

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

Alternative 8: 14.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in 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

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.4

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

7. Applied rewrites 14.4%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 14.4

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 14.4

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

   9. lift--.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-rgt-out-- N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. metadata-eval 14.4

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

9. Applied rewrites 14.4%

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

Alternative 9: 14.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -((single(1.0) / (((single(0.5) * single(pi)) * u) / s)) * 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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.4

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

7. Applied rewrites 14.4%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. distribute-lft-neg-out N/A

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

9. Applied rewrites 14.4%

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

Alternative 10: 11.5% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

4. Applied rewrites 11.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

6. Applied rewrites 11.5%

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

7. Taylor expanded in u around 0

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

   1. lower-+.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 11.5

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

9. Applied rewrites 11.5%

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

Alternative 11: 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 2025155 
(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))))