?

Average Accuracy: 98.5% → 98.7%
Time: 19.5s
Precision: binary32
Cost: 10208

?

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\frac{-0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}{-{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}} \]
(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
 (*
  (* cosTheta_i (/ cosTheta_O v))
  (/
   (/ (/ -0.5 v) (sinh (/ 1.0 v)))
   (- (pow (exp (/ sinTheta_i v)) sinTheta_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 (cosTheta_i * (cosTheta_O / v)) * (((-0.5f / v) / sinhf((1.0f / v))) / -powf(expf((sinTheta_i / v)), sinTheta_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 = (costheta_i * (costheta_o / v)) * ((((-0.5e0) / v) / sinh((1.0e0 / v))) / -(exp((sintheta_i / v)) ** sintheta_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(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(Float32(Float32(-0.5) / v) / sinh(Float32(Float32(1.0) / v))) / Float32(-(exp(Float32(sinTheta_i / v)) ^ sinTheta_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 = (cosTheta_i * (cosTheta_O / v)) * (((single(-0.5) / v) / sinh((single(1.0) / v))) / -(exp((sinTheta_i / v)) ^ sinTheta_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(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\frac{-0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}{-{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 98.5%

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

  2. Simplified98.5%

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

    [Start]98.5

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

    *-commutative [=>]98.5

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

    associate-*r/ [<=]98.5

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

    *-commutative [=>]98.5

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

    associate-/l* [=>]98.5

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

    associate-/r/ [=>]98.5

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

    *-commutative [=>]98.5

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

    associate-/r* [=>]98.5

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

  3. Applied egg-rr98.7%

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

  4. Simplified98.7%

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

    [Start]98.7

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

    associate-*l/ [=>]98.7

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

    associate-/r* [=>]98.7

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

    associate-*l/ [=>]98.7

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

    metadata-eval [=>]98.7

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

  5. Final simplification98.7%

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

Alternatives

Alternative 1
Accuracy98.7%
Cost10176
\[\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{v \cdot {\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\sinh \left(\frac{1}{v}\right)} \]
Alternative 2
Accuracy98.7%
Cost7104
\[\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot \frac{1}{v}\right)}{v} \]
Alternative 3
Accuracy98.5%
Cost7008
\[\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}{e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{v} \]
Alternative 4
Accuracy98.5%
Cost7008
\[\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}} \]
Alternative 5
Accuracy98.7%
Cost7008
\[\frac{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
Alternative 6
Accuracy98.3%
Cost3616
\[\frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]
Alternative 7
Accuracy98.4%
Cost3616
\[\frac{cosTheta_O}{2 \cdot \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)} \cdot \frac{cosTheta_i}{v} \]
Alternative 8
Accuracy98.4%
Cost3616
\[cosTheta_i \cdot \frac{cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \left(v \cdot v\right)\right)} \]
Alternative 9
Accuracy64.1%
Cost3552
\[cosTheta_O \cdot \left(cosTheta_i \cdot \frac{1}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}\right) \]
Alternative 10
Accuracy64.1%
Cost384
\[cosTheta_O \cdot \frac{-cosTheta_i}{v \cdot -2 + \frac{-0.3333333333333333}{v}} \]
Alternative 11
Accuracy64.1%
Cost352
\[\frac{cosTheta_i \cdot cosTheta_O}{v \cdot 2 + \frac{0.3333333333333333}{v}} \]
Alternative 12
Accuracy58.8%
Cost288
\[\frac{1}{\frac{v}{cosTheta_O \cdot \left(cosTheta_i \cdot 0.5\right)}} \]
Alternative 13
Accuracy58.3%
Cost224
\[0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \]
Alternative 14
Accuracy58.3%
Cost224
\[0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]
Alternative 15
Accuracy58.7%
Cost224
\[\frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \]

Error

Reproduce?

herbie shell --seed 2023129 
(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)))