Average Error: 0.5 → 0.4
Time: 9.8s
Precision: binary32
\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{1}{v}}\\ cosTheta_O \cdot \left(\frac{1}{v} \cdot \frac{cosTheta_i}{v \cdot \left(e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(t_0 + \frac{-1}{t_0}\right)\right)}\right) \end{array} \]
(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
 (let* ((t_0 (exp (/ 1.0 v))))
   (*
    cosTheta_O
    (*
     (/ 1.0 v)
     (/
      cosTheta_i
      (* v (* (exp (/ (* sinTheta_i sinTheta_O) v)) (+ t_0 (/ -1.0 t_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) {
	float t_0 = expf((1.0f / v));
	return cosTheta_O * ((1.0f / v) * (cosTheta_i / (v * (expf(((sinTheta_i * sinTheta_O) / v)) * (t_0 + (-1.0f / t_0))))));
}

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function
↓
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = exp((1.0e0 / v))
    code = costheta_o * ((1.0e0 / v) * (costheta_i / (v * (exp(((sintheta_i * sintheta_o) / v)) * (t_0 + ((-1.0e0) / t_0))))))
end function

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)
	t_0 = exp(Float32(Float32(1.0) / v))
	return Float32(cosTheta_O * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i / Float32(v * Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / v)) * Float32(t_0 + Float32(Float32(-1.0) / t_0)))))))
end

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end
↓
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = exp((single(1.0) / v));
	tmp = cosTheta_O * ((single(1.0) / v) * (cosTheta_i / (v * (exp(((sinTheta_i * sinTheta_O) / v)) * (t_0 + (single(-1.0) / t_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}
↓
\begin{array}{l}
t_0 := e^{\frac{1}{v}}\\
cosTheta_O \cdot \left(\frac{1}{v} \cdot \frac{cosTheta_i}{v \cdot \left(e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(t_0 + \frac{-1}{t_0}\right)\right)}\right)
\end{array}

Error

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. Simplified0.5

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

  3. Applied egg-rr0.4

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

  4. Taylor expanded in cosTheta_i around inf 0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022209 
(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)))