?

Average Accuracy: 99.7% → 99.5%
Time: 28.9s
Precision: binary32
Cost: 16320

?

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[{e}^{\left({\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}\right)}^{3}\right)} \]
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))
↓
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow E (pow (cbrt (+ (log (/ 0.5 v)) (+ 0.6931 (/ -1.0 v)))) 3.0)))
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}
↓
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(((float) M_E), powf(cbrtf((logf((0.5f / v)) + (0.6931f + (-1.0f / v)))), 3.0f));
}

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end
↓
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(1)) ^ (cbrt(Float32(log(Float32(Float32(0.5) / v)) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)))) ^ Float32(3.0))
end

e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}
↓
{e}^{\left({\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}\right)}^{3}\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.7%

    \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  2. Simplified99.7%

    \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
    Proof

    [Start]99.7

    \[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

    +-commutative [=>]99.7

    \[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

    log-div [=>]99.7

    \[ e^{\color{blue}{\left(\log 1 - \log \left(2 \cdot v\right)\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]

    metadata-eval [=>]99.7

    \[ e^{\left(\color{blue}{0} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]

    associate-+l- [=>]99.7

    \[ e^{\color{blue}{0 - \left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]

    associate-+l- [<=]99.7

    \[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

    metadata-eval [<=]99.7

    \[ e^{\left(\color{blue}{\log 1} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]

    log-div [<=]99.7

    \[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]

    +-commutative [<=]99.7

    \[ e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

    associate-+l+ [=>]99.7

    \[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

  3. Applied egg-rr99.6%

    \[\leadsto \color{blue}{{e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]
    Proof

    [Start]99.7

    \[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \]

    *-un-lft-identity [=>]99.7

    \[ e^{\color{blue}{1 \cdot \left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

    exp-prod [=>]99.6

    \[ \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

    exp-1-e [=>]99.6

    \[ {\color{blue}{e}}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

    associate-/r/ [=>]99.6

    \[ {e}^{\left(\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

    *-commutative [=>]99.6

    \[ {e}^{\left(\left(\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

    div-inv [=>]99.6

    \[ {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\color{blue}{sinTheta_i \cdot \frac{1}{\frac{v}{sinTheta_O}}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

    fma-def [=>]99.6

    \[ {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\mathsf{fma}\left(sinTheta_i, \frac{1}{\frac{v}{sinTheta_O}}, \frac{1}{v}\right)}\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

    clear-num [<=]99.6

    \[ {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \color{blue}{\frac{sinTheta_O}{v}}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

  4. Applied egg-rr99.6%

    \[\leadsto {e}^{\color{blue}{\left({\left(\sqrt[3]{cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\left(\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right) - 0.6931\right) - \log \left(\frac{0.5}{v}\right)\right)}\right)}^{3}\right)}} \]
    Proof

    [Start]99.6

    \[ {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

    add-cube-cbrt [=>]99.6

    \[ {e}^{\color{blue}{\left(\left(\sqrt[3]{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot \sqrt[3]{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot \sqrt[3]{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right)}} \]

    pow3 [=>]99.6

    \[ {e}^{\color{blue}{\left({\left(\sqrt[3]{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right)}^{3}\right)}} \]

    associate-+l- [=>]99.6

    \[ {e}^{\left({\left(\sqrt[3]{\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right) - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}}\right)}^{3}\right)} \]

    associate--r+ [=>]99.6

    \[ {e}^{\left({\left(\sqrt[3]{cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\left(\left(\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right) - 0.6931\right) - \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{3}\right)} \]

  5. Taylor expanded in cosTheta_O around 0 -0.0%

    \[\leadsto {e}^{\left({\color{blue}{\left({\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right)}^{0.3333333333333333}\right)}}^{3}\right)} \]

  6. Simplified99.6%

    \[\leadsto {e}^{\left({\color{blue}{\left(\sqrt[3]{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right) - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right)}}^{3}\right)} \]
    Proof

    [Start]-0.0

    \[ {e}^{\left({\left({\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right)}^{0.3333333333333333}\right)}^{3}\right)} \]

    unpow1/3 [=>]99.6

    \[ {e}^{\left({\color{blue}{\left(\sqrt[3]{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}\right)}}^{3}\right)} \]

    associate--r+ [=>]99.6

    \[ {e}^{\left({\left(\sqrt[3]{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right) - \frac{sinTheta_i \cdot sinTheta_O}{v}}}\right)}^{3}\right)} \]

    associate-/l* [=>]99.6

    \[ {e}^{\left({\left(\sqrt[3]{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right) - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}\right)}^{3}\right)} \]

  7. Taylor expanded in sinTheta_i around 0 -0.0%

    \[\leadsto {e}^{\left({\color{blue}{\left({\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right)}^{0.3333333333333333}\right)}}^{3}\right)} \]

  8. Simplified99.5%

    \[\leadsto {e}^{\left({\color{blue}{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right) + \left(0.6931 - \frac{1}{v}\right)}\right)}}^{3}\right)} \]
    Proof

    [Start]-0.0

    \[ {e}^{\left({\left({\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right)}^{0.3333333333333333}\right)}^{3}\right)} \]

    unpow1/3 [=>]99.5

    \[ {e}^{\left({\color{blue}{\left(\sqrt[3]{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}}\right)}}^{3}\right)} \]

    +-commutative [=>]99.5

    \[ {e}^{\left({\left(\sqrt[3]{\color{blue}{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)} - \frac{1}{v}}\right)}^{3}\right)} \]

    associate-+r- [<=]99.5

    \[ {e}^{\left({\left(\sqrt[3]{\color{blue}{\log \left(\frac{0.5}{v}\right) + \left(0.6931 - \frac{1}{v}\right)}}\right)}^{3}\right)} \]

  9. Final simplification99.5%

    \[\leadsto {e}^{\left({\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}\right)}^{3}\right)} \]

Alternatives

Alternative 1
Accuracy99.6%
Cost9888
\[{e}^{\left(\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) + \frac{-1}{v}\right)} \]
Alternative 2
Accuracy99.6%
Cost6688
\[e^{\log \left(\frac{0.5}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)} \]
Alternative 3
Accuracy99.4%
Cost4064
\[\begin{array}{l} t_0 := 0.6931 + \frac{1}{v}\\ \frac{0.5}{v} \cdot e^{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v} + t_0}{\frac{t_0}{0.6931 + \frac{-1}{v}}}} \end{array} \]
Alternative 4
Accuracy20.2%
Cost3588
\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 0:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{1}{\frac{v}{sinTheta_O}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{-v}}\\ \end{array} \]
Alternative 5
Accuracy20.2%
Cost3556
\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 0:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{1}{\frac{v}{sinTheta_O}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}\\ \end{array} \]
Alternative 6
Accuracy20.2%
Cost3524
\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 0:\\ \;\;\;\;e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}\\ \end{array} \]
Alternative 7
Accuracy99.6%
Cost3488
\[\frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
Alternative 8
Accuracy12.5%
Cost3360
\[e^{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]
Alternative 9
Accuracy12.5%
Cost3360
\[e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]
Alternative 10
Accuracy6.4%
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023136 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (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))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))