Average Error: 0.1 → 0.1
Time: 18.9s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := 0.6931 + \frac{cosTheta_i \cdot cosTheta_O + -1}{v}\\ t_1 := \frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\\ t_2 := \sqrt[3]{e^{t_1} \cdot \frac{0.5}{v}}\\ \left(t_2 \cdot t_2\right) \cdot \sqrt[3]{\frac{0.5}{v} \cdot {\left(e^{\sqrt[3]{{\left(\sqrt[3]{t_0}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{{t_0}^{4}}}}\right)}^{\left(\sqrt[3]{t_1}\right)}} \end{array} \]
(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 (+ 0.6931 (/ (+ (* cosTheta_i cosTheta_O) -1.0) v)))
        (t_1
         (+
          (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) v)
          (+ 0.6931 (/ -1.0 v))))
        (t_2 (cbrt (* (exp t_1) (/ 0.5 v)))))
   (*
    (* t_2 t_2)
    (cbrt
     (*
      (/ 0.5 v)
      (pow
       (exp (* (cbrt (pow (cbrt t_0) 2.0)) (cbrt (cbrt (pow t_0 4.0)))))
       (cbrt t_1)))))))
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 = 0.6931f + (((cosTheta_i * cosTheta_O) + -1.0f) / v);
	float t_1 = (((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + (0.6931f + (-1.0f / v));
	float t_2 = cbrtf((expf(t_1) * (0.5f / v)));
	return (t_2 * t_2) * cbrtf(((0.5f / v) * powf(expf((cbrtf(powf(cbrtf(t_0), 2.0f)) * cbrtf(cbrtf(powf(t_0, 4.0f))))), cbrtf(t_1))));
}

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(Float32(0.6931) + Float32(Float32(Float32(cosTheta_i * cosTheta_O) + Float32(-1.0)) / v))
	t_1 = Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) / v) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)))
	t_2 = cbrt(Float32(exp(t_1) * Float32(Float32(0.5) / v)))
	return Float32(Float32(t_2 * t_2) * cbrt(Float32(Float32(Float32(0.5) / v) * (exp(Float32(cbrt((cbrt(t_0) ^ Float32(2.0))) * cbrt(cbrt((t_0 ^ Float32(4.0)))))) ^ cbrt(t_1)))))
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 := 0.6931 + \frac{cosTheta_i \cdot cosTheta_O + -1}{v}\\
t_1 := \frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\\
t_2 := \sqrt[3]{e^{t_1} \cdot \frac{0.5}{v}}\\
\left(t_2 \cdot t_2\right) \cdot \sqrt[3]{\frac{0.5}{v} \cdot {\left(e^{\sqrt[3]{{\left(\sqrt[3]{t_0}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{{t_0}^{4}}}}\right)}^{\left(\sqrt[3]{t_1}\right)}}
\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Applied egg-rr0.1

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

  4. Taylor expanded in sinTheta_i around 0 0.1

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

  5. Applied egg-rr0.1

    \[\leadsto \left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right) \cdot \sqrt[3]{{\left(e^{\color{blue}{\sqrt[3]{{\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right)}^{4}}}}}\right)}^{\left(\sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} + -0.6931\right)}\right)} \cdot \frac{0.5}{v}} \]

  6. Final simplification0.1

    \[\leadsto \left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)} \cdot \frac{0.5}{v}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)} \cdot \frac{0.5}{v}}\right) \cdot \sqrt[3]{\frac{0.5}{v} \cdot {\left(e^{\sqrt[3]{{\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O + -1}{v}}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O + -1}{v}\right)}^{4}}}}\right)}^{\left(\sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)}\right)}} \]

Reproduce

herbie shell --seed 2022206 
(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))))))