?

Average Accuracy: 99.6% → 99.7%
Time: 22.1s
Precision: binary32
Cost: 6816

?

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[0.5 \cdot e^{\frac{1}{\frac{1}{\left(0.6931 - \frac{1}{v}\right) - \log v}}} \]
(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
 (* 0.5 (exp (/ 1.0 (/ 1.0 (- (- 0.6931 (/ 1.0 v)) (log v)))))))
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 0.5f * expf((1.0f / (1.0f / ((0.6931f - (1.0f / v)) - logf(v)))));
}

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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (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 * exp((1.0e0 / (1.0e0 / ((0.6931e0 - (1.0e0 / v)) - log(v)))))
end function

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(Float32(0.5) * exp(Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(Float32(Float32(0.6931) - Float32(Float32(1.0) / v)) - log(v))))))
end

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end
↓
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * exp((single(1.0) / (single(1.0) / ((single(0.6931) - (single(1.0) / v)) - log(v)))));
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)}
↓
0.5 \cdot e^{\frac{1}{\frac{1}{\left(0.6931 - \frac{1}{v}\right) - \log v}}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.6%

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

  2. Simplified99.7%

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

    [Start]99.6

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

    remove-double-neg [<=]99.6

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

    +-commutative [<=]99.6

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

    log-rec [=>]99.8

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

    distribute-neg-in [<=]99.8

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

    sub-neg [<=]99.8

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

    exp-neg [=>]99.7

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

    exp-diff [=>]99.7

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

  3. Applied egg-rr99.7%

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

    [Start]99.7

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

    add-exp-log [=>]99.7

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

    div-exp [=>]99.7

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

    associate--l- [=>]99.7

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

  4. Taylor expanded in sinTheta_O around 0 99.7%

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

  5. Applied egg-rr99.7%

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

    [Start]99.7

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

    flip-- [=>]97.3

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

    clear-num [=>]97.3

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

    *-un-lft-identity [=>]97.3

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

    associate-/l* [=>]97.3

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

    flip-- [<=]99.7

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

    associate--l+ [=>]99.7

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

    associate-/l* [=>]99.7

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

  6. Taylor expanded in cosTheta_i around 0 99.7%

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

  7. Simplified99.7%

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

    [Start]99.7

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

    associate--r+ [=>]99.7

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

    sub-neg [=>]99.7

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

    distribute-neg-frac [=>]99.7

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

    metadata-eval [=>]99.7

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

  8. Final simplification99.7%

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

Alternatives

Alternative 1
Accuracy99.7%
Cost6688
\[0.5 \cdot e^{\left(0.6931 - \frac{1}{v}\right) - \log v} \]
Alternative 2
Accuracy99.7%
Cost3488
\[0.5 \cdot \frac{e^{0.6931 - \frac{1}{v}}}{v} \]
Alternative 3
Accuracy12.8%
Cost3424
\[0.5 \cdot e^{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]
Alternative 4
Accuracy97.9%
Cost3424
\[e^{\frac{-1}{v}} \cdot \frac{0.5}{v} \]
Alternative 5
Accuracy98.1%
Cost3424
\[\frac{\frac{e^{\frac{-1}{v}}}{v}}{2} \]
Alternative 6
Accuracy4.7%
Cost96
\[\frac{0.5}{v} \]

Error

Reproduce?

herbie shell --seed 2023139 
(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))))))