Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 25.5s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = ((u / (single(-1.0) - t_0)) + (u / (single(1.0) + exp((single(pi) / -s))))) + (single(1.0) / (single(1.0) + t_0));
	tmp = s * log((((t_1 ^ single(-2.0)) + (single(1.0) + (single(1.0) / t_1))) / (single(-1.0) + (t_1 ^ single(-3.0)))));
end

Derivation

1. Initial program 98.8%

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

   \[\leadsto s \cdot \log \left(\frac{{\left(\left(\frac{u}{-1 - e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 + \frac{1}{\left(\frac{u}{-1 - e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\left(\frac{u}{-1 - e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right) \]
8. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. flip-+ N/A

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. clear-num N/A

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

   2. /-lowering-/.f32 N/A

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

   3. /-lowering-/.f32 N/A

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

   4. PI-lowering-PI.f32 98.8

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

4. Applied egg-rr 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

4. Applied egg-rr 98.8%

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

5. Final simplification 98.8%

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

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 99.0%

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

8. Applied egg-rr 98.9%

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

10. Applied egg-rr 98.8%

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

11. Final simplification 98.8%

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

Alternative 6: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 99.0%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in u around inf

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. +-lowering-+.f32 N/A

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Alternative 7: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in u around inf

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

   1. *-lowering-*.f32 N/A

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

   2. sub-neg N/A

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

   3. +-lowering-+.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. +-lowering-+.f32 N/A

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

   6. exp-lowering-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. mul-1-neg N/A

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

   10. /-lowering-/.f32 N/A

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

   11. PI-lowering-PI.f32 N/A

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

   12. mul-1-neg N/A

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

   13. neg-lowering-neg.f32 N/A

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

   14. distribute-neg-frac N/A

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

5. Simplified 97.3%

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

6. Final simplification 97.3%

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

Alternative 8: 25.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 24.4%

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

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

8. Simplified 24.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, 2 \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} + \frac{u \cdot \left(\pi \cdot \pi\right)}{s \cdot \left(\left(1 + \frac{\pi}{s}\right) \cdot \left(1 + \frac{\pi}{s}\right)\right)}\right), \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]

9. Taylor expanded in s around 0

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

    1. *-lowering-*.f32 N/A

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

    2. +-commutative N/A

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

    3. accelerator-lowering-fma.f32 N/A

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

    4. *-lowering-*.f32 N/A

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

    5. +-lowering-+.f32 N/A

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

    6. mul-1-neg N/A

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

    7. neg-lowering-neg.f32 N/A

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

    8. +-lowering-+.f32 N/A

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

    9. log-lowering-log.f32 N/A

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

    10. PI-lowering-PI.f32 N/A

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

    11. mul-1-neg N/A

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

    12. neg-lowering-neg.f32 N/A

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

    13. log-lowering-log.f32 24.9

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

11. Simplified 24.9%

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

12. Final simplification 24.9%

    \[\leadsto s \cdot \mathsf{fma}\left(2, u \cdot \left(u + 1\right), \log s - \log \pi\right) \]
13. Add Preprocessing

Alternative 9: 25.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 24.4%

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

6. Taylor expanded in u around 0

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

   1. associate-*r* N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. mul-1-neg N/A

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

   4. neg-lowering-neg.f32 N/A

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

   5. accelerator-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \color{blue}{\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}, 2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right), 2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s}\right), 2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\frac{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\frac{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{\color{blue}{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u\right)}}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u\right)}}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   13. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot u\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   14. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)}{\color{blue}{1 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)}{1 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}\right) \]

   16. PI-lowering-PI.f32 24.8

       \[\leadsto \mathsf{fma}\left(-s, \mathsf{log1p}\left(\frac{\pi}{s}\right), \frac{2 \cdot \left(\pi \cdot u\right)}{1 + \frac{\color{blue}{\pi}}{s}}\right) \]

8. Simplified 24.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-s, \mathsf{log1p}\left(\frac{\pi}{s}\right), \frac{2 \cdot \left(\pi \cdot u\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0

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

    1. *-lowering-*.f32 N/A

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

    2. +-commutative N/A

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

    3. mul-1-neg N/A

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

    4. unsub-neg N/A

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

    5. --lowering--.f32 N/A

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

    6. *-commutative N/A

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

    7. *-lowering-*.f32 N/A

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

    8. mul-1-neg N/A

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

    9. unsub-neg N/A

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

    10. --lowering--.f32 N/A

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

    11. log-lowering-log.f32 N/A

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

    12. PI-lowering-PI.f32 N/A

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

    13. log-lowering-log.f32 24.9

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

11. Simplified 24.9%

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

12. Final simplification 24.9%

    \[\leadsto s \cdot \left(u \cdot 2 + \left(\log s - \log \pi\right)\right) \]
13. Add Preprocessing

Alternative 10: 25.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \left(u + 1\right) \cdot \left(s \cdot 2\right), \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma u (* (+ u 1.0) (* s 2.0)) (* (- s) (log1p (/ PI s)))))

C

float code(float u, float s) {
	return fmaf(u, ((u + 1.0f) * (s * 2.0f)), (-s * log1pf((((float) M_PI) / s))));
}

Julia

function code(u, s)
	return fma(u, Float32(Float32(u + Float32(1.0)) * Float32(s * Float32(2.0))), Float32(Float32(-s) * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 24.4%

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

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

8. Simplified 24.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, 2 \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} + \frac{u \cdot \left(\pi \cdot \pi\right)}{s \cdot \left(\left(1 + \frac{\pi}{s}\right) \cdot \left(1 + \frac{\pi}{s}\right)\right)}\right), \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]

9. Taylor expanded in s around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 \cdot \left(s \cdot \left(1 + u\right)\right)}, \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(2 \cdot s\right) \cdot \left(1 + u\right)}, \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(2 \cdot s\right) \cdot \left(1 + u\right)}, \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(2 \cdot s\right)} \cdot \left(1 + u\right), \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]

    4. +-lowering-+.f32 24.8

       \[\leadsto \mathsf{fma}\left(u, \left(2 \cdot s\right) \cdot \color{blue}{\left(1 + u\right)}, \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]

11. Simplified 24.8%

    \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(2 \cdot s\right) \cdot \left(1 + u\right)}, \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]

12. Final simplification 24.8%

    \[\leadsto \mathsf{fma}\left(u, \left(u + 1\right) \cdot \left(s \cdot 2\right), \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]
13. Add Preprocessing

Alternative 11: 25.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-s, \mathsf{log1p}\left(\frac{\pi}{s}\right), 2 \cdot \left(s \cdot u\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (fma (- s) (log1p (/ PI s)) (* 2.0 (* s u))))

C

float code(float u, float s) {
	return fmaf(-s, log1pf((((float) M_PI) / s)), (2.0f * (s * u)));
}

Julia

function code(u, s)
	return fma(Float32(-s), log1p(Float32(Float32(pi) / s)), Float32(Float32(2.0) * Float32(s * u)))
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 24.4%

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

6. Taylor expanded in u around 0

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

   1. associate-*r* N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. mul-1-neg N/A

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

   4. neg-lowering-neg.f32 N/A

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

   5. accelerator-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \color{blue}{\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}, 2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right), 2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s}\right), 2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\frac{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\frac{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{\color{blue}{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u\right)}}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u\right)}}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   13. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot u\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   14. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)}{\color{blue}{1 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)}{1 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}\right) \]

   16. PI-lowering-PI.f32 24.8

       \[\leadsto \mathsf{fma}\left(-s, \mathsf{log1p}\left(\frac{\pi}{s}\right), \frac{2 \cdot \left(\pi \cdot u\right)}{1 + \frac{\color{blue}{\pi}}{s}}\right) \]

8. Simplified 24.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-s, \mathsf{log1p}\left(\frac{\pi}{s}\right), \frac{2 \cdot \left(\pi \cdot u\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{2 \cdot \left(s \cdot u\right)}\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(s\right), \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{2 \cdot \left(s \cdot u\right)}\right) \]

    2. *-lowering-*.f32 24.8

       \[\leadsto \mathsf{fma}\left(-s, \mathsf{log1p}\left(\frac{\pi}{s}\right), 2 \cdot \color{blue}{\left(s \cdot u\right)}\right) \]

11. Simplified 24.8%

    \[\leadsto \mathsf{fma}\left(-s, \mathsf{log1p}\left(\frac{\pi}{s}\right), \color{blue}{2 \cdot \left(s \cdot u\right)}\right) \]
12. Add Preprocessing

Alternative 12: 25.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 24.4%

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

6. Taylor expanded in u around 0

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

   1. +-lowering-+.f32 N/A

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

   2. /-lowering-/.f32 N/A

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

   3. PI-lowering-PI.f32 24.8

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

8. Simplified 24.8%

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

Alternative 13: 25.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 24.4%

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

6. Taylor expanded in u around 0

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

   1. accelerator-lowering-log1p.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   3. PI-lowering-PI.f32 24.8

      \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{\pi}}{s}\right) \]

8. Simplified 24.8%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]
9. Add Preprocessing

Alternative 14: 11.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot \left(\pi \cdot 0.5\right)\\ t_1 := \left(u \cdot \pi\right) \cdot \left(u \cdot \pi\right)\\ \frac{\mathsf{fma}\left(t\_0, t\_1 \cdot 0.25, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.015625\right) \cdot 4}{\mathsf{fma}\left(t\_1, 0.25, \left(\pi \cdot -0.25\right) \cdot \left(\pi \cdot -0.25 - t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* u (* PI 0.5))) (t_1 (* (* u PI) (* u PI))))
   (/
    (* (fma t_0 (* t_1 0.25) (* (* PI (* PI PI)) -0.015625)) 4.0)
    (fma t_1 0.25 (* (* PI -0.25) (- (* PI -0.25) t_0))))))

C

float code(float u, float s) {
	float t_0 = u * (((float) M_PI) * 0.5f);
	float t_1 = (u * ((float) M_PI)) * (u * ((float) M_PI));
	return (fmaf(t_0, (t_1 * 0.25f), ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * -0.015625f)) * 4.0f) / fmaf(t_1, 0.25f, ((((float) M_PI) * -0.25f) * ((((float) M_PI) * -0.25f) - t_0)));
}

Julia

function code(u, s)
	t_0 = Float32(u * Float32(Float32(pi) * Float32(0.5)))
	t_1 = Float32(Float32(u * Float32(pi)) * Float32(u * Float32(pi)))
	return Float32(Float32(fma(t_0, Float32(t_1 * Float32(0.25)), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(-0.015625))) * Float32(4.0)) / fma(t_1, Float32(0.25), Float32(Float32(Float32(pi) * Float32(-0.25)) * Float32(Float32(Float32(pi) * Float32(-0.25)) - t_0))))
end

Derivation

1. Initial program 98.8%

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

   1. flip-- N/A

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

   2. div-inv N/A

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in s around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{-16}{3} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-4}{3} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{-16}{3} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-4}{3} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-16}{3} - \frac{-4}{3}\right)}\right) \]

   3. metadata-eval N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. metadata-eval N/A

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

   6. *-lowering-*.f32 N/A

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

7. Simplified 11.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \pi \cdot 0.5, \pi \cdot -0.25\right) \cdot 4} \]
8. Step-by-step derivation

   1. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{\left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right)}^{3} + {\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)}^{3}}{\left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) \cdot \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) + \left(\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) - \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)\right)}} \cdot 4 \]

   2. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left({\left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right)}^{3} + {\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)}^{3}\right) \cdot 4}{\left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) \cdot \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) + \left(\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) - \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\left({\left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right)}^{3} + {\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)}^{3}\right) \cdot 4}{\left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) \cdot \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) + \left(\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) - \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)\right)}} \]

9. Applied egg-rr 11.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u \cdot \left(\pi \cdot 0.5\right), \left(\left(u \cdot \pi\right) \cdot \left(u \cdot \pi\right)\right) \cdot 0.25, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.015625\right) \cdot 4}{\mathsf{fma}\left(\left(u \cdot \pi\right) \cdot \left(u \cdot \pi\right), 0.25, \left(\pi \cdot -0.25\right) \cdot \left(\pi \cdot -0.25 - u \cdot \left(\pi \cdot 0.5\right)\right)\right)}} \]
10. Add Preprocessing

Alternative 15: 11.5% accurate, 36.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

   1. flip-- N/A

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

   2. div-inv N/A

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in s around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{-16}{3} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-4}{3} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{-16}{3} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-4}{3} \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-16}{3} - \frac{-4}{3}\right)}\right) \]

   3. metadata-eval N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. metadata-eval N/A

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

   6. *-lowering-*.f32 N/A

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

7. Simplified 11.4%

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

8. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

   6. PI-lowering-PI.f32 N/A

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

   7. mul-1-neg N/A

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

   8. neg-lowering-neg.f32 N/A

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

   9. PI-lowering-PI.f32 11.4

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

10. Simplified 11.4%

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

11. Final simplification 11.4%

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

Alternative 16: 11.3% accurate, 170.0× 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 98.8%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f32 N/A

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

   3. PI-lowering-PI.f32 11.2

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

5. Simplified 11.2%

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

Alternative 17: 10.3% accurate, 510.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 98.8%

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in s around inf

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

9. Simplified 10.4%

   \[\leadsto \color{blue}{s \cdot 0} \]
10. Step-by-step derivation

    1. mul0-rgt 10.4

       \[\leadsto \color{blue}{0} \]

11. Applied egg-rr 10.4%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))