?

Average Error: 1.45% → 1.31%
Time: 17.1s
Precision: binary32
Cost: 7104

?

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]
\[ \begin{array}{c}[cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \end{array} \]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]
(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
 (*
  (/ (exp (/ (- sinTheta_i) (/ v sinTheta_O))) (* (sinh (/ 1.0 v)) 2.0))
  (* (/ 1.0 v) (* (/ cosTheta_i 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 (expf((-sinTheta_i / (v / sinTheta_O))) / (sinhf((1.0f / v)) * 2.0f)) * ((1.0f / v) * ((cosTheta_i / 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 = (exp((-sintheta_i / (v / sintheta_o))) / (sinh((1.0e0 / v)) * 2.0e0)) * ((1.0e0 / v) * ((costheta_i / 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(exp(Float32(Float32(-sinTheta_i) / Float32(v / sinTheta_O))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) * Float32(Float32(Float32(1.0) / v) * Float32(Float32(cosTheta_i / 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 = (exp((-sinTheta_i / (v / sinTheta_O))) / (sinh((single(1.0) / v)) * single(2.0))) * ((single(1.0) / v) * ((cosTheta_i / 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}
↓
\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 1.45

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

  2. Simplified1.5

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

    [Start]1.45

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

    times-frac [=>]1.5

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

    associate-/l* [=>]1.5

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

  3. Applied egg-rr1.31

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

  4. Final simplification1.31

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

Alternatives

Alternative 1
Error1.39%
Cost7040
\[cosTheta_O \cdot \frac{\frac{-cosTheta_i}{\left(v \cdot v\right) \cdot \left(e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \cdot -2\right)}}{\sinh \left(\frac{1}{v}\right)} \]
Alternative 2
Error1.61%
Cost7008
\[\left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
Alternative 3
Error1.41%
Cost7008
\[\frac{\frac{\frac{cosTheta_i}{v}}{v \cdot 2}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
Alternative 4
Error1.39%
Cost7008
\[cosTheta_O \cdot \frac{\frac{\frac{cosTheta_i}{v \cdot v}}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \cdot 0.5}{\sinh \left(\frac{1}{v}\right)} \]
Alternative 5
Error1.55%
Cost6944
\[\left(\frac{1}{v} \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
Alternative 6
Error1.71%
Cost6880
\[cosTheta_O \cdot \frac{cosTheta_i}{\left(v \cdot v\right) \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)} \]
Alternative 7
Error1.78%
Cost3616
\[\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
Alternative 8
Error1.69%
Cost3616
\[cosTheta_O \cdot \frac{\frac{cosTheta_i \cdot 0.5}{v \cdot v}}{\sinh \left(\frac{1}{v}\right)} \]
Alternative 9
Error41.17%
Cost480
\[\left(v \cdot 0.5\right) \cdot \left(\frac{1}{v} \cdot \frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}}\right) \]
Alternative 10
Error41.17%
Cost352
\[\frac{v \cdot 0.5}{v \cdot \frac{\frac{v}{cosTheta_i}}{cosTheta_O}} \]
Alternative 11
Error41.64%
Cost224
\[0.5 \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \]
Alternative 12
Error41.64%
Cost224
\[\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right) \cdot 0.5 \]
Alternative 13
Error41.63%
Cost224
\[0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}} \]
Alternative 14
Error41.63%
Cost224
\[0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]
Alternative 15
Error41.17%
Cost224
\[\frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \]

Error

Reproduce?

herbie shell --seed 2023089 
(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)))