HairBSDF, Mp, upper

Average Error: 0.5 → 0.5

Time: 17.5s

Precision: binary32

Cost: 40512

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{1}{v}}\\ t_1 := t_0 - \frac{1}{t_0}\\ t_2 := t_1 \cdot t_1\\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{v \cdot \left(t_1 \cdot \left(\left|\frac{1}{t_2}\right| \cdot t_2\right)\right)} \end{array} \]

FPCore

(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
 (let* ((t_0 (exp (/ 1.0 v))) (t_1 (- t_0 (/ 1.0 t_0))) (t_2 (* t_1 t_1)))
   (/
    (*
     (exp (- (/ (* sinTheta_i sinTheta_O) v)))
     (/ (* cosTheta_i cosTheta_O) v))
    (* v (* t_1 (* (fabs (/ 1.0 t_2)) t_2))))))

C

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) {
	float t_0 = expf((1.0f / v));
	float t_1 = t_0 - (1.0f / t_0);
	float t_2 = t_1 * t_1;
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / (v * (t_1 * (fabsf((1.0f / t_2)) * t_2)));
}

Fortran

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
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: t_2
    t_0 = exp((1.0e0 / v))
    t_1 = t_0 - (1.0e0 / t_0)
    t_2 = t_1 * t_1
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / (v * (t_1 * (abs((1.0e0 / t_2)) * t_2)))
end function

Julia

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)
	t_0 = exp(Float32(Float32(1.0) / v))
	t_1 = Float32(t_0 - Float32(Float32(1.0) / t_0))
	t_2 = Float32(t_1 * t_1)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(v * Float32(t_1 * Float32(abs(Float32(Float32(1.0) / t_2)) * t_2))))
end

MATLAB

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)
	t_0 = exp((single(1.0) / v));
	t_1 = t_0 - (single(1.0) / t_0);
	t_2 = t_1 * t_1;
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / (v * (t_1 * (abs((single(1.0) / t_2)) * t_2)));
end

Derivation

1. Initial program 0.5

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

2. Taylor expanded in v around 0 0.5

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

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.5

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

5. Simplified 0.5

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

   [Start] 0.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{v \cdot \left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot \left|\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot \left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right)\right) \cdot \frac{1}{\left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right) \cdot \left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right)}\right|\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{v \cdot \left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot \left|\color{blue}{\frac{1}{\left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right) \cdot \left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right)} \cdot \left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot \left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right)\right)}\right|\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-72 [=>] 0.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{v \cdot \left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot \color{blue}{\left(\left|\frac{1}{\left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right) \cdot \left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right)}\right| \cdot \left|\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot \left(\frac{1}{e^{\frac{1}{v}}} - e^{\frac{1}{v}}\right)\right|\right)}\right)} \]

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	20480

\[\begin{array}{l} t_0 := \sinh \left(\frac{1}{v}\right)\\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\left(\left(t_0 \cdot \frac{1}{\left(t_0 \cdot \frac{1}{t_0}\right) \cdot t_0}\right) \cdot t_0\right) \cdot 2\right) \cdot v} \end{array} \]

Alternative 2
Error	0.5
Cost	7040

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

Alternative 3
Error	11.6
Cost	7008

\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{0.3333333333333333}{{v}^{2}}} \]

Alternative 4
Error	11.6
Cost	3648

\[\frac{cosTheta_i \cdot cosTheta_O}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)} \]

Alternative 5
Error	13.5
Cost	224

\[0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]

Reproduce

herbie shell --seed 2023090 
(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)))