Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 1.1min

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (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

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

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (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: 97.7% accurate, 1.4× speedup

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

Math

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

FPCore

(FPCore (u s)
  :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 (- (exp (- (/ PI s))) -1.0))
      (/ 1.0 (- (exp (/ PI s)) -1.0)))))
   1.0))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

Alternative 2: 94.2% accurate, 1.6× speedup

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

Math

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

FPCore

(FPCore (u s)
  :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 (- (exp (- (/ PI s))) -1.0))
      (/ 1.0 (+ 2.0 (/ PI s))))))
   1.0))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 94.2%

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

9. Applied rewrites 94.2%

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

Alternative 3: 36.2% accurate, 2.0× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{1}{2 + \frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - t\_0\right) + t\_0} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (let* ((t_0 (/ 1.0 (+ 2.0 (/ PI s)))))
  (* (- s) (log (- (/ 1.0 (+ (* u (- 0.5 t_0)) t_0)) 1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (2.0f + (((float) M_PI) / s));
	return -s * logf(((1.0f / ((u * (0.5f - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s)))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(0.5) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(2.0) + (single(pi) / s));
	tmp = -s * log(((single(1.0) / ((u * (single(0.5) - t_0)) + t_0)) - single(1.0)));
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 85.8%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 36.2%

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

Alternative 4: 25.1% accurate, 4.6× speedup

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

Math

\[\left(-s\right) \cdot \log \left(\mathsf{fma}\left(\pi, \frac{1}{s}, 1\right)\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (fma PI (/ 1.0 s) 1.0))))

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(fma(Float32(pi), Float32(Float32(1.0) / 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. Taylor expanded in s around inf

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

4. Applied rewrites 24.9%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 25.1%

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

9. Applied rewrites 25.1%

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

Alternative 5: 25.1% accurate, 5.2× speedup

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

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- 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 98.9%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 24.9%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 25.1%

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

Alternative 6: 25.1% accurate, 6.1× speedup

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

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (/ PI s))))

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((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

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

4. Applied rewrites 24.9%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 25.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 24.9%

    \[\leadsto \left(-s\right) \cdot \log \left(-4 \cdot \frac{u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi}{s}\right) \]

11. Taylor expanded in u around 0

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

13. Applied rewrites 25.1%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{\pi}{s}\right) \]
14. Add Preprocessing

Alternative 7: 14.4% accurate, 6.2× speedup

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

Math

\[-1 \cdot \frac{s}{u \cdot \frac{1.5707963705062866}{s}} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* -1.0 (/ s (* u (/ 1.5707963705062866 s)))))

C

float code(float u, float s) {
	return -1.0f * (s / (u * (1.5707963705062866f / s)));
}

Fortran

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = (-1.0e0) * (s / (u * (1.5707963705062866e0 / s)))
end function

Julia

function code(u, s)
	return Float32(Float32(-1.0) * Float32(s / Float32(u * Float32(Float32(1.5707963705062866) / s))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(-1.0) * (s / (u * (single(1.5707963705062866) / 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

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 14.4%

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

8. Evaluated real constant 14.4%

   \[\leadsto -1 \cdot \frac{s}{u \cdot \left(-1 \cdot \frac{-1.5707963705062866}{s}\right)} \]

10. Applied rewrites 14.4%

    \[\leadsto -1 \cdot \frac{s}{u \cdot \frac{1.5707963705062866}{s}} \]
11. Add Preprocessing

Alternative 8: 14.4% accurate, 6.2× speedup

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

Math

\[-1 \cdot \frac{s}{1.5707963705062866 \cdot \frac{u}{s}} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* -1.0 (/ s (* 1.5707963705062866 (/ u s)))))

C

float code(float u, float s) {
	return -1.0f * (s / (1.5707963705062866f * (u / s)));
}

Fortran

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = (-1.0e0) * (s / (1.5707963705062866e0 * (u / s)))
end function

Julia

function code(u, s)
	return Float32(Float32(-1.0) * Float32(s / Float32(Float32(1.5707963705062866) * Float32(u / s))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(-1.0) * (s / (single(1.5707963705062866) * (u / 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

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 14.4%

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

8. Evaluated real constant 14.4%

   \[\leadsto -1 \cdot \frac{s}{-1 \cdot \frac{u \cdot -1.5707963705062866}{s}} \]

9. Taylor expanded in u around 0

   \[\leadsto -1 \cdot \frac{s}{\frac{13176795}{8388608} \cdot \frac{u}{s}} \]

11. Applied rewrites 14.4%

    \[\leadsto -1 \cdot \frac{s}{1.5707963705062866 \cdot \frac{u}{s}} \]
12. Add Preprocessing

Alternative 9: 11.6% accurate, 7.9× speedup

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

Math

\[-u \cdot \left(\frac{\pi}{u} - 6.2831854820251465\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (- (* u (- (/ PI u) 6.2831854820251465))))

C

float code(float u, float s) {
	return -(u * ((((float) M_PI) / u) - 6.2831854820251465f));
}

Julia

function code(u, s)
	return Float32(-Float32(u * Float32(Float32(Float32(pi) / u) - Float32(6.2831854820251465))))
end

MATLAB

function tmp = code(u, s)
	tmp = -(u * ((single(pi) / u) - single(6.2831854820251465)));
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

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

4. Applied rewrites 11.6%

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

5. Evaluated real constant 11.6%

   \[\leadsto -4 \cdot \left(u \cdot -1.5707963705062866 - -0.25 \cdot \pi\right) \]

6. Taylor expanded in u around -inf

   \[\leadsto -1 \cdot \left(u \cdot \left(\frac{\pi}{u} - \frac{13176795}{2097152}\right)\right) \]

8. Applied rewrites 11.6%

   \[\leadsto -1 \cdot \left(u \cdot \left(\frac{\pi}{u} - 6.2831854820251465\right)\right) \]

10. Applied rewrites 11.6%

    \[\leadsto -u \cdot \left(\frac{\pi}{u} - 6.2831854820251465\right) \]
11. Add Preprocessing

Alternative 10: 11.6% accurate, 11.9× speedup

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

Math

\[\left(-\pi\right) + 6.2831854820251465 \cdot u \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (+ (- PI) (* 6.2831854820251465 u)))

C

float code(float u, float s) {
	return -((float) M_PI) + (6.2831854820251465f * u);
}

Julia

function code(u, s)
	return Float32(Float32(-Float32(pi)) + Float32(Float32(6.2831854820251465) * u))
end

MATLAB

function tmp = code(u, s)
	tmp = -single(pi) + (single(6.2831854820251465) * u);
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around -inf

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

4. Applied rewrites 11.6%

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

5. Evaluated real constant 11.6%

   \[\leadsto -4 \cdot \left(u \cdot -1.5707963705062866 - -0.25 \cdot \pi\right) \]

6. Taylor expanded in u around 0

   \[\leadsto -1 \cdot \pi + \frac{13176795}{2097152} \cdot u \]

8. Applied rewrites 11.6%

   \[\leadsto \mathsf{fma}\left(-1, \pi, 6.2831854820251465 \cdot u\right) \]

10. Applied rewrites 11.6%

    \[\leadsto \left(-\pi\right) + 6.2831854820251465 \cdot u \]
11. Add Preprocessing

Alternative 11: 11.6% accurate, 12.9× speedup

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

Math

\[\mathsf{fma}\left(6.2831854820251465, u, -\pi\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (fma 6.2831854820251465 u (- PI)))

C

float code(float u, float s) {
	return fmaf(6.2831854820251465f, u, -((float) M_PI));
}

Julia

function code(u, s)
	return fma(Float32(6.2831854820251465), u, Float32(-Float32(pi)))
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around -inf

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

4. Applied rewrites 11.6%

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

5. Evaluated real constant 11.6%

   \[\leadsto -4 \cdot \left(u \cdot -1.5707963705062866 - -0.25 \cdot \pi\right) \]

6. Taylor expanded in u around 0

   \[\leadsto -1 \cdot \pi + \frac{13176795}{2097152} \cdot u \]

8. Applied rewrites 11.6%

   \[\leadsto \mathsf{fma}\left(-1, \pi, 6.2831854820251465 \cdot u\right) \]

10. Applied rewrites 11.6%

    \[\leadsto \mathsf{fma}\left(6.2831854820251465, u, -\pi\right) \]
11. Add Preprocessing

Alternative 12: 11.3% accurate, 90.2× speedup

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

Math

\[-3.1415927410125732 \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  -3.1415927410125732)

C

float code(float u, float s) {
	return -3.1415927410125732f;
}

Fortran

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -3.1415927410125732e0
end function

Julia

function code(u, s)
	return Float32(-3.1415927410125732)
end

MATLAB

function tmp = code(u, s)
	tmp = single(-3.1415927410125732);
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

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

4. Applied rewrites 11.3%

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

5. Evaluated real constant 11.3%

   \[\leadsto -3.1415927410125732 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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))))