?

Average Accuracy: 98.6% → 98.8%
Time: 21.5s
Precision: binary32
Cost: 7072

?

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]
\[ \begin{array}{c}[cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \end{array} \]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[\left(\frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)} \cdot cosTheta_i\right) \cdot \left(\frac{1}{v} \cdot cosTheta_O\right) \]
(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
 (*
  (*
   (/ 0.5 (* (sinh (/ 1.0 v)) (* v (exp (/ (* sinTheta_i sinTheta_O) v)))))
   cosTheta_i)
  (* (/ 1.0 v) cosTheta_O)))
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 ((0.5f / (sinhf((1.0f / v)) * (v * expf(((sinTheta_i * sinTheta_O) / v))))) * cosTheta_i) * ((1.0f / v) * cosTheta_O);
}

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
    code = ((0.5e0 / (sinh((1.0e0 / v)) * (v * exp(((sintheta_i * sintheta_o) / v))))) * costheta_i) * ((1.0e0 / v) * costheta_o)
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)
	return Float32(Float32(Float32(Float32(0.5) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * exp(Float32(Float32(sinTheta_i * sinTheta_O) / v))))) * cosTheta_i) * Float32(Float32(Float32(1.0) / v) * cosTheta_O))
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)
	tmp = ((single(0.5) / (sinh((single(1.0) / v)) * (v * exp(((sinTheta_i * sinTheta_O) / v))))) * cosTheta_i) * ((single(1.0) / v) * cosTheta_O);
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}
↓
\left(\frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)} \cdot cosTheta_i\right) \cdot \left(\frac{1}{v} \cdot cosTheta_O\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 98.6%

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

  2. Simplified98.6%

    \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot {\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v}} \]
    Proof

    [Start]98.6

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

    *-commutative [=>]98.6

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

    associate-*r/ [<=]98.6

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

    *-commutative [=>]98.6

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

    associate-/l* [=>]98.6

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

    associate-/r/ [=>]98.6

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

    *-commutative [=>]98.6

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

    associate-/r* [=>]98.6

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

  3. Applied egg-rr98.2%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot {\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}\right)} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v}} \]
    Proof
  4. No proof available- proof too large to flatten.

  5. Applied egg-rr98.8%

    \[\leadsto \color{blue}{\left(\frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot \left(e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot v\right)} \cdot cosTheta_i\right) \cdot \left(cosTheta_O \cdot \frac{1}{v}\right)} \]
    Proof
  6. No proof available- proof too large to flatten.

  7. Final simplification98.8%

    \[\leadsto \left(\frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)} \cdot cosTheta_i\right) \cdot \left(\frac{1}{v} \cdot cosTheta_O\right) \]

Alternatives

Alternative 1
Accuracy98.6%
Cost7008
\[cosTheta_i \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot \left(\frac{v}{cosTheta_O} \cdot \left(v \cdot e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right)\right)} \]
Alternative 2
Accuracy98.8%
Cost7008
\[\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)} \cdot \left(cosTheta_i \cdot cosTheta_O\right) \]
Alternative 3
Accuracy98.4%
Cost6880
\[\frac{cosTheta_O}{v \cdot v} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
Alternative 4
Accuracy98.3%
Cost6880
\[\frac{cosTheta_i}{\frac{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}{\frac{cosTheta_O}{v \cdot v}}} \]
Alternative 5
Accuracy98.3%
Cost6880
\[\frac{\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O}}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
Alternative 6
Accuracy98.3%
Cost3616
\[\frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v \cdot v} \]
Alternative 7
Accuracy68.5%
Cost3552
\[cosTheta_i \cdot \frac{cosTheta_O}{\left(v \cdot v\right) \cdot \mathsf{expm1}\left(\frac{1}{v}\right)} \]
Alternative 8
Accuracy68.5%
Cost3552
\[\frac{cosTheta_O}{v \cdot v} \cdot \frac{cosTheta_i}{\mathsf{expm1}\left(\frac{1}{v}\right)} \]
Alternative 9
Accuracy68.5%
Cost3552
\[\frac{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{v}}{\mathsf{expm1}\left(\frac{1}{v}\right)} \]
Alternative 10
Accuracy59.3%
Cost288
\[\frac{1}{\frac{\frac{v}{0.5 \cdot cosTheta_i}}{cosTheta_O}} \]
Alternative 11
Accuracy59.3%
Cost288
\[\frac{\frac{0.5}{v}}{\frac{\frac{1}{cosTheta_O}}{cosTheta_i}} \]
Alternative 12
Accuracy58.7%
Cost224
\[0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]
Alternative 13
Accuracy58.7%
Cost224
\[cosTheta_O \cdot \left(0.5 \cdot \frac{cosTheta_i}{v}\right) \]
Alternative 14
Accuracy58.7%
Cost224
\[cosTheta_i \cdot \frac{0.5}{\frac{v}{cosTheta_O}} \]
Alternative 15
Accuracy58.7%
Cost224
\[\frac{cosTheta_O}{v} \cdot \left(0.5 \cdot cosTheta_i\right) \]
Alternative 16
Accuracy59.2%
Cost224
\[\frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \]
Alternative 17
Accuracy59.2%
Cost224
\[\frac{0.5}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \]
Alternative 18
Accuracy56.0%
Cost160
\[cosTheta_O \cdot \frac{cosTheta_i}{v} \]
Alternative 19
Accuracy56.0%
Cost160
\[\frac{cosTheta_i \cdot cosTheta_O}{v} \]

Error

Reproduce?

herbie shell --seed 2023133 
(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)))