Average Error: 0.1 → 0.1
Time: 11.0s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{-\log \left(v \cdot 2\right)}\\ e^{\mathsf{fma}\left(t_0, t_0, \frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right)} \end{array} \]
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))
↓
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (* v 2.0))))))
   (exp
    (fma
     t_0
     t_0
     (+
      (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) v)
      (+ 0.6931 (/ -1.0 v)))))))
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}
↓
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = sqrtf(-logf((v * 2.0f)));
	return expf(fmaf(t_0, t_0, ((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + (0.6931f + (-1.0f / v)))));
}

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end
↓
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = sqrt(Float32(-log(Float32(v * Float32(2.0)))))
	return exp(fma(t_0, t_0, Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) / v) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)))))
end

e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}
↓
\begin{array}{l}
t_0 := \sqrt{-\log \left(v \cdot 2\right)}\\
e^{\mathsf{fma}\left(t_0, t_0, \frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right)}
\end{array}

Error

Derivation

  1. Initial program 0.1

    \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  2. Applied egg-rr0.1

    \[\leadsto e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{\left(-\log \left(v \cdot 2\right)\right)}} \]

  3. Applied egg-rr0.1

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\sqrt{-\log \left(v \cdot 2\right)}, \sqrt{-\log \left(v \cdot 2\right)}, \frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right)}} \]

  4. Final simplification0.1

    \[\leadsto e^{\mathsf{fma}\left(\sqrt{-\log \left(v \cdot 2\right)}, \sqrt{-\log \left(v \cdot 2\right)}, \frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right)} \]

Reproduce

herbie shell --seed 2022211 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))