Average Error: 0.1 → 0.2
Time: 11.2s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\\ {\left(\left(\sqrt[3]{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot t_0\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{t_0}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}}}\right)}^{3} \cdot \frac{0.5}{v} \end{array} \]
(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
         (-
          (fma cosTheta_O (/ cosTheta_i v) 0.6931)
          (fma sinTheta_i (/ sinTheta_O v) (/ 1.0 v)))))
   (*
    (pow
     (*
      (*
       (cbrt (exp (* 0.6666666666666666 (* 0.3333333333333333 t_0))))
       (cbrt (cbrt (pow (cbrt (exp t_0)) 4.0))))
      (cbrt (cbrt (exp (+ 0.6931 (/ (fma cosTheta_i cosTheta_O -1.0) v))))))
     3.0)
    (/ 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) {
	float t_0 = fmaf(cosTheta_O, (cosTheta_i / v), 0.6931f) - fmaf(sinTheta_i, (sinTheta_O / v), (1.0f / v));
	return powf(((cbrtf(expf((0.6666666666666666f * (0.3333333333333333f * t_0)))) * cbrtf(cbrtf(powf(cbrtf(expf(t_0)), 4.0f)))) * cbrtf(cbrtf(expf((0.6931f + (fmaf(cosTheta_i, cosTheta_O, -1.0f) / v)))))), 3.0f) * (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)
	t_0 = Float32(fma(cosTheta_O, Float32(cosTheta_i / v), Float32(0.6931)) - fma(sinTheta_i, Float32(sinTheta_O / v), Float32(Float32(1.0) / v)))
	return Float32((Float32(Float32(cbrt(exp(Float32(Float32(0.6666666666666666) * Float32(Float32(0.3333333333333333) * t_0)))) * cbrt(cbrt((cbrt(exp(t_0)) ^ Float32(4.0))))) * cbrt(cbrt(exp(Float32(Float32(0.6931) + Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v)))))) ^ Float32(3.0)) * 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)}
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\\
{\left(\left(\sqrt[3]{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot t_0\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{t_0}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}}}\right)}^{3} \cdot \frac{0.5}{v}
\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. Simplified0.1

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

    \[\leadsto {\color{blue}{\left({\left({\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}}\right)}}^{3} \cdot \frac{0.5}{v} \]

  5. Applied egg-rr0.2

    \[\leadsto {\left(\color{blue}{\left(\sqrt[3]{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{4}}}\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}}\right)}^{3} \cdot \frac{0.5}{v} \]

  6. Taylor expanded in sinTheta_i around 0 0.2

    \[\leadsto {\left(\left(\sqrt[3]{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\color{blue}{{\left(e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}\right)}^{0.3333333333333333}}}\right)}^{3} \cdot \frac{0.5}{v} \]

  7. Simplified0.2

    \[\leadsto {\left(\left(\sqrt[3]{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\color{blue}{\sqrt[3]{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}}}}\right)}^{3} \cdot \frac{0.5}{v} \]

  8. Final simplification0.2

    \[\leadsto {\left(\left(\sqrt[3]{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{4}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}}}\right)}^{3} \cdot \frac{0.5}{v} \]

Reproduce

herbie shell --seed 2022185 
(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))))))