Average Error: 0.5 → 0.4
Time: 7.2s
Precision: binary32
\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]
\[ \begin{array}{c}[cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \end{array} \]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[\frac{e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}} \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{1}{v}\right)\right)}{\sqrt[3]{{\left(v \cdot 2\right)}^{3} \cdot {\sinh \left(\frac{1}{v}\right)}^{3}}} \]
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))
↓
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i (- sinTheta_O)) v))
   (* cosTheta_i (* cosTheta_O (/ 1.0 v))))
  (cbrt (* (pow (* v 2.0) 3.0) (pow (sinh (/ 1.0 v)) 3.0)))))
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}
↓
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_i * -sinTheta_O) / v)) * (cosTheta_i * (cosTheta_O * (1.0f / v)))) / cbrtf((powf((v * 2.0f), 3.0f) * powf(sinhf((1.0f / v)), 3.0f)));
}

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end
↓
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v)) * Float32(cosTheta_i * Float32(cosTheta_O * Float32(Float32(1.0) / v)))) / cbrt(Float32((Float32(v * Float32(2.0)) ^ Float32(3.0)) * (sinh(Float32(Float32(1.0) / v)) ^ Float32(3.0)))))
end

\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}
↓
\frac{e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}} \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{1}{v}\right)\right)}{\sqrt[3]{{\left(v \cdot 2\right)}^{3} \cdot {\sinh \left(\frac{1}{v}\right)}^{3}}}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Applied egg-rr0.4

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

  3. Applied egg-rr0.4

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

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022153 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))