HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.6%

Time: 15.6s

Alternatives: 9

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

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)))

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);
}

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

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

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

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)))

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);
}

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

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

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

Alternative 1: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i sinTheta_O) (- v)))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (* v (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_i * sinTheta_O) / -v)) * ((1.0f / v) * (cosTheta_i * cosTheta_O))) / (v * (expf((1.0f / v)) - expf((-1.0f / v))));
}

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)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / (v * (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((sinTheta_i * sinTheta_O) / -v)) * ((single(1.0) / v) * (cosTheta_i * cosTheta_O))) / (v * (exp((single(1.0) / v)) - exp((single(-1.0) / v))));
end

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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.8%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Taylor expanded in v around 0 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\color{blue}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
6. Step-by-step derivation

   1. rec-exp 98.9%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{v \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

   2. distribute-neg-frac 98.9%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{v \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

   3. metadata-eval 98.9%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

7. Simplified 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\color{blue}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

8. Final simplification 98.9%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)} \]
9. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i sinTheta_O) (- v)))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((sinTheta_i * sinTheta_O) / -v)) * ((1.0f / v) * (cosTheta_i * cosTheta_O))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

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)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((sinTheta_i * sinTheta_O) / -v)) * ((single(1.0) / v) * (cosTheta_i * cosTheta_O))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.8%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Final simplification 98.8%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
6. Add Preprocessing

Alternative 3: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* (/ 1.0 v) (* cosTheta_i cosTheta_O)) (* v (* (sinh (/ 1.0 v)) 2.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((1.0f / v) * (cosTheta_i * cosTheta_O)) / (v * (sinhf((1.0f / v)) * 2.0f));
}

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 = ((1.0e0 / v) * (costheta_i * costheta_o)) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O)) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(1.0) / v) * (cosTheta_i * cosTheta_O)) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.8%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. distribute-neg-frac2 98.8%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. add-sqr-sqrt -0.0%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. sqrt-unprod 98.8%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. sqr-neg 98.8%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\sqrt{\color{blue}{v \cdot v}}}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. sqrt-unprod 98.8%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. add-sqr-sqrt 98.8%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{v}}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

6. Applied egg-rr 98.8%

   \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

7. Taylor expanded in sinTheta_i around 0 98.8%

   \[\leadsto \frac{\color{blue}{1} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

8. Final simplification 98.8%

   \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
9. Add Preprocessing

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ cosTheta\_i \cdot \frac{0.5 \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (/ (* 0.5 (/ (/ cosTheta_O v) (sinh (/ 1.0 v)))) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i * ((0.5f * ((cosTheta_O / v) / sinhf((1.0f / v)))) / v);
}

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 = costheta_i * ((0.5e0 * ((costheta_o / v) / sinh((1.0e0 / v)))) / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * Float32(Float32(Float32(0.5) * Float32(Float32(cosTheta_O / v) / sinh(Float32(Float32(1.0) / v)))) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * ((single(0.5) * ((cosTheta_O / v) / sinh((single(1.0) / v)))) / v);
end

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. Step-by-step derivation

   1. associate-*r/ 98.6%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-/l/ 98.7%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   3. remove-double-neg 98.7%

      \[\leadsto \frac{\color{blue}{-\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   4. distribute-rgt-neg-out 98.7%

      \[\leadsto \frac{-\color{blue}{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(-cosTheta\_i \cdot cosTheta\_O\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   5. distribute-rgt-neg-out 98.7%

      \[\leadsto \frac{-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_i \cdot \left(-cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   6. distribute-lft-neg-in 98.7%

      \[\leadsto \frac{\color{blue}{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \left(cosTheta\_i \cdot \left(-cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   7. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \color{blue}{\frac{\frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   9. associate-*r/ 98.6%

      \[\leadsto \color{blue}{\frac{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow-exp 98.6%

      \[\leadsto \frac{\color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   2. times-frac 98.5%

      \[\leadsto \color{blue}{\frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. add-sqr-sqrt 80.1%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. sqrt-unprod 98.5%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. sqr-neg 98.5%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\sqrt{\color{blue}{v \cdot v}}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. sqrt-unprod 98.5%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. add-sqr-sqrt 98.5%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\color{blue}{v}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

7. Taylor expanded in sinTheta_i around 0 98.5%

   \[\leadsto \color{blue}{\frac{1}{v}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
8. Step-by-step derivation

   1. associate-*l/ 98.3%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{v}} \]

   2. *-un-lft-identity 98.3%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}}}{v} \]

   3. associate-/l* 98.2%

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}}}{v} \]

   4. *-un-lft-identity 98.2%

      \[\leadsto \frac{cosTheta\_i \cdot \frac{\color{blue}{1 \cdot \frac{cosTheta\_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{v} \]

   5. *-commutative 98.2%

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1 \cdot \frac{cosTheta\_O}{v}}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}}}{v} \]

   6. times-frac 98.2%

      \[\leadsto \frac{cosTheta\_i \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}\right)}}{v} \]

   7. metadata-eval 98.2%

      \[\leadsto \frac{cosTheta\_i \cdot \left(\color{blue}{0.5} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}\right)}{v} \]

9. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(0.5 \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}\right)}{v}} \]
10. Step-by-step derivation

    1. associate-/l* 98.4%

       \[\leadsto \color{blue}{cosTheta\_i \cdot \frac{0.5 \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}}{v}} \]

11. Simplified 98.4%

    \[\leadsto \color{blue}{cosTheta\_i \cdot \frac{0.5 \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}}{v}} \]
12. Add Preprocessing

Alternative 5: 57.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot \frac{cosTheta\_O \cdot \left(\left(sinTheta\_i \cdot sinTheta\_O\right) \cdot cosTheta\_i\right)}{v} + 0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (+
   (* -0.5 (/ (* cosTheta_O (* (* sinTheta_i sinTheta_O) cosTheta_i)) v))
   (* 0.5 (* cosTheta_i cosTheta_O)))
  v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((-0.5f * ((cosTheta_O * ((sinTheta_i * sinTheta_O) * cosTheta_i)) / v)) + (0.5f * (cosTheta_i * cosTheta_O))) / v;
}

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 = (((-0.5e0) * ((costheta_o * ((sintheta_i * sintheta_o) * costheta_i)) / v)) + (0.5e0 * (costheta_i * costheta_o))) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(cosTheta_O * Float32(Float32(sinTheta_i * sinTheta_O) * cosTheta_i)) / v)) + Float32(Float32(0.5) * Float32(cosTheta_i * cosTheta_O))) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(-0.5) * ((cosTheta_O * ((sinTheta_i * sinTheta_O) * cosTheta_i)) / v)) + (single(0.5) * (cosTheta_i * cosTheta_O))) / v;
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 57.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + 0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

6. Final simplification 57.7%

   \[\leadsto \frac{-0.5 \cdot \frac{cosTheta\_O \cdot \left(\left(sinTheta\_i \cdot sinTheta\_O\right) \cdot cosTheta\_i\right)}{v} + 0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v} \]
7. Add Preprocessing

Alternative 6: 57.7% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{v} \cdot \left(0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 1.0 v) (* 0.5 (* cosTheta_i cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (1.0f / v) * (0.5f * (cosTheta_i * cosTheta_O));
}

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 = (1.0e0 / v) * (0.5e0 * (costheta_i * costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(1.0) / v) * Float32(Float32(0.5) * Float32(cosTheta_i * cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) / v) * (single(0.5) * (cosTheta_i * cosTheta_O));
end

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. Step-by-step derivation

   1. associate-*r/ 98.6%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-/l/ 98.7%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   3. remove-double-neg 98.7%

      \[\leadsto \frac{\color{blue}{-\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   4. distribute-rgt-neg-out 98.7%

      \[\leadsto \frac{-\color{blue}{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(-cosTheta\_i \cdot cosTheta\_O\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   5. distribute-rgt-neg-out 98.7%

      \[\leadsto \frac{-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_i \cdot \left(-cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   6. distribute-lft-neg-in 98.7%

      \[\leadsto \frac{\color{blue}{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \left(cosTheta\_i \cdot \left(-cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   7. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \color{blue}{\frac{\frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   9. associate-*r/ 98.6%

      \[\leadsto \color{blue}{\frac{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow-exp 98.6%

      \[\leadsto \frac{\color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   2. times-frac 98.5%

      \[\leadsto \color{blue}{\frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. add-sqr-sqrt 80.1%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. sqrt-unprod 98.5%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. sqr-neg 98.5%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\sqrt{\color{blue}{v \cdot v}}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. sqrt-unprod 98.5%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. add-sqr-sqrt 98.5%

      \[\leadsto \frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{\color{blue}{v}}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}{v} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

7. Taylor expanded in sinTheta_i around 0 98.5%

   \[\leadsto \color{blue}{\frac{1}{v}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

8. Taylor expanded in v around inf 57.7%

   \[\leadsto \frac{1}{v} \cdot \color{blue}{\left(0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 57.7%

      \[\leadsto \frac{1}{v} \cdot \color{blue}{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5\right)} \]

10. Simplified 57.7%

    \[\leadsto \frac{1}{v} \cdot \color{blue}{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5\right)} \]

11. Final simplification 57.7%

    \[\leadsto \frac{1}{v} \cdot \left(0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right) \]
12. Add Preprocessing

Alternative 7: 57.7% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{1}{v}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* cosTheta_O (* cosTheta_i (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_O * (cosTheta_i * (1.0f / v)));
}

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 = 0.5e0 * (costheta_o * (costheta_i * (1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(cosTheta_O * Float32(cosTheta_i * Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_O * (cosTheta_i * (single(1.0) / v)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 57.6%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

6. Taylor expanded in v around inf 57.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. associate-/l* 57.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

8. Simplified 57.6%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]
9. Step-by-step derivation

   1. div-inv 57.6%

      \[\leadsto 0.5 \cdot \left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{1}{v}\right)}\right) \]

10. Applied egg-rr 57.6%

    \[\leadsto 0.5 \cdot \left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{1}{v}\right)}\right) \]
11. Add Preprocessing

Alternative 8: 57.7% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* cosTheta_O (/ cosTheta_i v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_O * (cosTheta_i / v));
}

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 = 0.5e0 * (costheta_o * (costheta_i / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(cosTheta_O * Float32(cosTheta_i / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_O * (cosTheta_i / v));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 57.6%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

6. Taylor expanded in v around inf 57.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. associate-/l* 57.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

8. Simplified 57.6%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]
9. Add Preprocessing

Alternative 9: 51.6% accurate, 220.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 0.0)

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.0f;
}

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 = 0.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(0.0)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
end

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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.8%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Taylor expanded in v around 0 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\color{blue}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
6. Step-by-step derivation

   1. rec-exp 98.9%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{v \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

   2. distribute-neg-frac 98.9%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{v \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

   3. metadata-eval 98.9%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

7. Simplified 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}{\color{blue}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

9. Applied egg-rr 2.5%

   \[\leadsto \color{blue}{1 \cdot \left({\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{1}{v}}\right)}\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 2.5%

       \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{1}{v}}\right)}} \]

    2. associate-*r/ 2.5%

       \[\leadsto \color{blue}{\frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)} \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{1}{v}}\right)}} \]

    3. *-commutative 2.5%

       \[\leadsto \frac{\color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{1}{v}}\right)} \]

    4. *-commutative 2.5%

       \[\leadsto \frac{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{1}{v}}\right) \cdot v}} \]

    5. *-lft-identity 2.5%

       \[\leadsto \frac{\color{blue}{\left(1 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}{\left(e^{\frac{1}{v}} - e^{\frac{1}{v}}\right) \cdot v} \]

    6. associate-*l* 2.5%

       \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}}{\left(e^{\frac{1}{v}} - e^{\frac{1}{v}}\right) \cdot v} \]

    7. +-inverses 2.5%

       \[\leadsto \frac{1 \cdot \left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}{\color{blue}{0} \cdot v} \]

    8. mul0-lft 2.5%

       \[\leadsto \frac{1 \cdot \left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}{\color{blue}{0}} \]

    9. +-inverses 2.5%

       \[\leadsto \frac{1 \cdot \left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}{\color{blue}{e^{\frac{1}{v}} - e^{\frac{1}{v}}}} \]

    10. associate-*l/ 2.5%

        \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - e^{\frac{1}{v}}} \cdot \left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)} \]

    11. associate-/r/ 2.5%

        \[\leadsto \color{blue}{\frac{1}{\frac{e^{\frac{1}{v}} - e^{\frac{1}{v}}}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}}} \]

11. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{1}{0}} \]
12. Step-by-step derivation

    1. inv-pow 3.2%

       \[\leadsto \color{blue}{{0}^{-1}} \]

    2. pow-base-0 50.9%

       \[\leadsto \color{blue}{0} \]

13. Applied egg-rr 50.9%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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)))