Average Error: 0.1 → 0.1
Time: 1.6min
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[e^{\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}\right)}^{2}\right)\right), \sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}}, 0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \]
(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
 (exp
  (fma
   (log1p
    (expm1
     (pow
      (cbrt
       (/ (- (* cosTheta_i cosTheta_O) (fma sinTheta_i sinTheta_O 1.0)) v))
      2.0)))
   (cbrt
    (+
     (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) v)
     (/ -1.0 v)))
   (+ 0.6931 (log (/ 0.5 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) {
	return expf(fmaf(log1pf(expm1f(powf(cbrtf((((cosTheta_i * cosTheta_O) - fmaf(sinTheta_i, sinTheta_O, 1.0f)) / v)), 2.0f))), cbrtf(((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + (-1.0f / v))), (0.6931f + logf((0.5f / 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)
	return exp(fma(log1p(expm1((cbrt(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - fma(sinTheta_i, sinTheta_O, Float32(1.0))) / v)) ^ Float32(2.0)))), cbrt(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) / v) + Float32(Float32(-1.0) / v))), Float32(Float32(0.6931) + log(Float32(Float32(0.5) / 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)}
↓
e^{\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}\right)}^{2}\right)\right), \sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}}, 0.6931 + \log \left(\frac{0.5}{v}\right)\right)}

Error

Bits error versus cosTheta_i

Bits error versus cosTheta_O

Bits error versus sinTheta_i

Bits error versus sinTheta_O

Bits error versus v

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

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}}\right)}^{2}, \sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}}, 0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

  3. Applied egg-rr0.1

    \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v}}\right)}^{2}\right)\right)}, \sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}}, 0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \]

  4. Applied egg-rr0.1

    \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}\right)}^{2}\right)\right)}, \sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}}, 0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \]

  5. Final simplification0.1

    \[\leadsto e^{\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}\right)}^{2}\right)\right), \sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}}, 0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \]

Reproduce

herbie shell --seed 2022148 
(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))))))