HairBSDF, Mp, upper

?

Percentage Accurate: 98.6% → 98.7%
Time: 19.8s
Precision: binary32
Cost: 7008

?

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

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

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

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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%

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

    [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

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

    *-commutative [=>]98.6

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

    *-commutative [=>]98.6

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

    associate-*r* [=>]98.6

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

    associate-/l/ [<=]98.6

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

    exp-neg [=>]98.6

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

    associate-/l/ [=>]98.6

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

    associate-/r* [=>]98.6

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

    metadata-eval [=>]98.6

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

    associate-*l/ [<=]98.6

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

    *-commutative [=>]98.6

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

    exp-prod [=>]98.6

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

  3. Applied egg-rr98.8%

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

    [Start]98.6

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

    clear-num [=>]98.6

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

    associate-/r/ [=>]98.6

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

    associate-/r* [=>]98.8

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

  4. Applied egg-rr98.7%

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

    [Start]98.8

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

    *-commutative [=>]98.8

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

    frac-times [=>]98.8

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

    associate-*l/ [=>]98.7

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

    associate-*l/ [=>]98.7

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

    metadata-eval [=>]98.7

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

    pow-exp [=>]98.7

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

    clear-num [=>]98.7

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

    un-div-inv [=>]98.7

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

  5. Applied egg-rr98.9%

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

    [Start]98.7

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

    associate-*r* [=>]98.9

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

    clear-num [=>]98.9

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

    un-div-inv [=>]98.9

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

  6. Applied egg-rr98.7%

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

    [Start]98.9

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

    associate-/l* [=>]93.5

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

    associate-/r/ [=>]98.7

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

  7. Final simplification98.7%

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

Alternatives

Alternative 1
Accuracy98.5%
Cost6944
\[\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
Alternative 2
Accuracy98.4%
Cost6880
\[\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta_i}{v \cdot v} \]
Alternative 3
Accuracy98.4%
Cost6880
\[\frac{\frac{cosTheta_i}{v}}{v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
Alternative 4
Accuracy97.5%
Cost3616
\[\frac{\frac{cosTheta_O \cdot cosTheta_i}{\frac{\sinh \left(\frac{1}{v}\right)}{0.5}}}{v \cdot v} \]
Alternative 5
Accuracy63.9%
Cost416
\[\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
Alternative 6
Accuracy63.9%
Cost416
\[\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
Alternative 7
Accuracy63.9%
Cost416
\[\frac{\frac{cosTheta_O \cdot cosTheta_i}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
Alternative 8
Accuracy58.8%
Cost288
\[\frac{1}{\frac{\frac{v}{cosTheta_O}}{0.5 \cdot cosTheta_i}} \]
Alternative 9
Accuracy58.2%
Cost224
\[0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}} \]
Alternative 10
Accuracy58.2%
Cost224
\[0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}} \]
Alternative 11
Accuracy58.2%
Cost224
\[0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v} \]
Alternative 12
Accuracy58.7%
Cost224
\[\frac{0.5}{\frac{v}{cosTheta_O \cdot cosTheta_i}} \]

Error

Reproduce?

herbie shell --seed 2023160 
(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)))