HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 100.0%

Time: 24.2s

Alternatives: 17

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: 98.6% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (/
    (exp (/ sinTheta_i (/ v (- sinTheta_O))))
    (/ (* v (* (sinh (/ 1.0 v)) 2.0)) (/ cosTheta_i (/ v cosTheta_O)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	double tmp = exp((((double) sinTheta_i) / (((double) v) / -((double) sinTheta_O)))) / ((((double) v) * (sinh((1.0 / ((double) v))) * 2.0)) / (((double) cosTheta_i) / (((double) v) / ((double) cosTheta_O))));
	return (float) tmp;
}

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
    real(8) :: tmp
    tmp = exp((real(sintheta_i, 8) / (real(v, 8) / -real(sintheta_o, 8)))) / ((real(v, 8) * (sinh((1.0d0 / real(v, 8))) * 2.0d0)) / (real(costheta_i, 8) / (real(v, 8) / real(costheta_o, 8))))
    code = real(tmp, 4)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float64(exp(Float64(Float64(sinTheta_i) / Float64(Float64(v) / Float64(-Float64(sinTheta_O))))) / Float64(Float64(Float64(v) * Float64(sinh(Float64(1.0 / Float64(v))) * 2.0)) / Float64(Float64(cosTheta_i) / Float64(Float64(v) / Float64(cosTheta_O)))))
	return Float32(tmp)
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((double(sinTheta_i) / (double(v) / -sinTheta_O))) / ((double(v) * (sinh((1.0 / double(v))) * 2.0)) / (double(cosTheta_i) / (double(v) / double(cosTheta_O))));
	tmp_2 = single(tmp);
end

Derivation

1. Initial program 100.0%

   \[\langle \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \rangle_{\text{binary64}} \]

2. Final simplification 100.0%

   \[\leadsto \langle \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \rangle_{\text{binary64}} \]

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \left(2 \cdot \langle \left( \sinh \left(\frac{1}{v}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ (* cosTheta_i cosTheta_O) (* v (exp (/ (* sinTheta_i sinTheta_O) v))))
  (* v (* 2.0 (cast (! :precision binary64 (sinh (/ 1.0 v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	double tmp = sinh((1.0 / ((double) v)));
	return ((cosTheta_i * cosTheta_O) / (v * expf(((sinTheta_i * sinTheta_O) / v)))) / (v * (2.0f * ((float) tmp)));
}

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
    real(8) :: tmp
    tmp = sinh((1.0d0 / real(v, 8)))
    code = ((costheta_i * costheta_o) / (v * exp(((sintheta_i * sintheta_o) / v)))) / (v * (2.0e0 * real(tmp, 4)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sinh(Float64(1.0 / Float64(v)))
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / Float32(v * exp(Float32(Float32(sinTheta_i * sinTheta_O) / v)))) / Float32(v * Float32(Float32(2.0) * Float32(tmp))))
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sinh((1.0 / double(v)));
	tmp_2 = ((cosTheta_i * cosTheta_O) / (v * exp(((sinTheta_i * sinTheta_O) / v)))) / (v * single((double(single(2.0)) * single(tmp))));
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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_O around 0 98.6%

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

   1. rewrite-binary32/binary64 99.1%

      \[\leadsto \color{blue}{\frac{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{v \cdot \left(\langle \sinh \left(\frac{1}{v}\right) \rangle_{\text{binary64}} \cdot 2\right)}} \]

6. Applied rewrite-once 99.1%

   \[\leadsto \frac{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{v \cdot \left(\color{blue}{\langle \color{blue}{\sinh \left(\frac{1}{v}\right)} \rangle_{\text{binary64}}} \cdot 2\right)} \]

7. Final simplification 99.1%

   \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \left(2 \cdot \langle \sinh \left(\frac{1}{v}\right) \rangle_{\text{binary64}}\right)} \]

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}{v \cdot \left(2 \cdot \langle \left( \sinh \left(\frac{1}{v}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ (/ cosTheta_i (/ v cosTheta_O)) (exp (* sinTheta_O (/ sinTheta_i v))))
  (* v (* 2.0 (cast (! :precision binary64 (sinh (/ 1.0 v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	double tmp = sinh((1.0 / ((double) v)));
	return ((cosTheta_i / (v / cosTheta_O)) / expf((sinTheta_O * (sinTheta_i / v)))) / (v * (2.0f * ((float) tmp)));
}

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
    real(8) :: tmp
    tmp = sinh((1.0d0 / real(v, 8)))
    code = ((costheta_i / (v / costheta_o)) / exp((sintheta_o * (sintheta_i / v)))) / (v * (2.0e0 * real(tmp, 4)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sinh(Float64(1.0 / Float64(v)))
	return Float32(Float32(Float32(cosTheta_i / Float32(v / cosTheta_O)) / exp(Float32(sinTheta_O * Float32(sinTheta_i / v)))) / Float32(v * Float32(Float32(2.0) * Float32(tmp))))
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sinh((1.0 / double(v)));
	tmp_2 = ((cosTheta_i / (v / cosTheta_O)) / exp((sinTheta_O * (sinTheta_i / v)))) / (v * single((double(single(2.0)) * single(tmp))));
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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_O around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/l* 98.6%

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

   4. associate-*r/ 98.6%

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

6. Simplified 98.6%

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

   1. rewrite-binary32/binary64 99.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}{v \cdot \left(\langle \sinh \left(\frac{1}{v}\right) \rangle_{\text{binary64}} \cdot 2\right)}} \]

8. Applied rewrite-once 99.2%

   \[\leadsto \frac{\frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}{v \cdot \left(\color{blue}{\langle \color{blue}{\sinh \left(\frac{1}{v}\right)} \rangle_{\text{binary64}}} \cdot 2\right)} \]

9. Final simplification 99.2%

   \[\leadsto \frac{\frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}{v \cdot \left(2 \cdot \langle \sinh \left(\frac{1}{v}\right) \rangle_{\text{binary64}}\right)} \]

Alternative 4: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * (cosTheta_i * (1.0f / v))) / expf((sinTheta_O * (sinTheta_i / v)))) / (v * (2.0f * sinhf((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 = ((costheta_o * (costheta_i * (1.0e0 / v))) / exp((sintheta_o * (sintheta_i / v)))) / (v * (2.0e0 * sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O * (cosTheta_i * (single(1.0) / v))) / exp((sinTheta_O * (sinTheta_i / v)))) / (v * (single(2.0) * sinh((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. Step-by-step derivation

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_O around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/l* 98.6%

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

   4. associate-*r/ 98.6%

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

6. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

   2. *-commutative 98.6%

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

   3. div-inv 98.9%

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

8. Applied egg-rr 98.9%

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

   1. associate-*l* 98.8%

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

10. Simplified 98.8%

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

11. Final simplification 98.8%

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

Alternative 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) * (1.0f / v)) / expf((sinTheta_O * (sinTheta_i / v)))) / (v * (2.0f * sinhf((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 = (((costheta_i * costheta_o) * (1.0e0 / v)) / exp((sintheta_o * (sintheta_i / v)))) / (v * (2.0e0 * sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) * (single(1.0) / v)) / exp((sinTheta_O * (sinTheta_i / v)))) / (v * (single(2.0) * sinh((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. Step-by-step derivation

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_O around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/l* 98.6%

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

   4. associate-*r/ 98.6%

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

6. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

   2. *-commutative 98.6%

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

   3. div-inv 98.9%

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

   4. *-commutative 98.9%

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 6: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / (v * expf(((sinTheta_i * sinTheta_O) / v)))) / (v * (2.0f * sinhf((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 = ((costheta_i * costheta_o) / (v * exp(((sintheta_i * sintheta_o) / v)))) / (v * (2.0e0 * sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / (v * exp(((sinTheta_i * sinTheta_O) / v)))) / (v * (single(2.0) * sinh((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. Step-by-step derivation

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_O around 0 98.6%

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

5. Final simplification 98.6%

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

Alternative 7: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i / (v / cosTheta_O)) / expf((sinTheta_O * (sinTheta_i / v)))) / (v * (2.0f * sinhf((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 = ((costheta_i / (v / costheta_o)) / exp((sintheta_o * (sintheta_i / v)))) / (v * (2.0e0 * sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i / (v / cosTheta_O)) / exp((sinTheta_O * (sinTheta_i / v)))) / (v * (single(2.0) * sinh((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. Step-by-step derivation

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_O around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/l* 98.6%

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

   4. associate-*r/ 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 8: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((1.0f / v) / (expf((1.0f / v)) - expf((-1.0f / v)))) * (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 = ((1.0e0 / v) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * (costheta_o * (costheta_i / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 93.9%

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

   2. *-commutative 93.9%

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

   3. distribute-neg-frac 93.9%

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

   4. distribute-rgt-neg-out 93.9%

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

   5. associate-/l* 93.9%

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

   6. *-commutative 93.9%

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

   7. *-commutative 93.9%

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

   8. associate-/l* 93.9%

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

3. Simplified 93.9%

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

   1. associate-/l* 93.9%

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

   2. *-commutative 93.9%

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

   3. associate-*l/ 93.9%

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

   4. associate-/l* 98.6%

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

   5. associate-/r/ 98.6%

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

   6. frac-times 98.5%

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

   7. *-commutative 98.5%

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

   8. div-inv 98.4%

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

   9. associate-*l* 98.7%

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

   10. *-commutative 98.7%

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

5. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(\frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{-v}\right)}}{v}\right)} \]
6. Step-by-step derivation

   1. add-exp-log_binary32 63.0%

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

7. Applied rewrite-once 63.0%

   \[\leadsto \color{blue}{e^{\log \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(\frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{-v}\right)}}{v}\right)\right)}} \]
8. Step-by-step derivation

   1. rem-exp-log 98.7%

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

   2. associate-*r/ 98.8%

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

   3. *-commutative 98.8%

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

   4. associate-*r/ 98.7%

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

9. Simplified 98.7%

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

10. Taylor expanded in sinTheta_i around 0 98.3%

    \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \color{blue}{\frac{1}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
11. Step-by-step derivation

    1. associate-/r* 98.5%

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

    2. rec-exp 98.5%

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

    3. distribute-neg-frac 98.5%

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

    4. metadata-eval 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 9: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i (/ cosTheta_O v))
  (/ (/ 1.0 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 (cosTheta_i * (cosTheta_O / v)) * ((1.0f / 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 = (costheta_i * (costheta_o / v)) * ((1.0e0 / 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(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(Float32(1.0) / 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 = (cosTheta_i * (cosTheta_O / v)) * ((single(1.0) / 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. Step-by-step derivation

   1. associate-/l* 93.9%

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

   2. *-commutative 93.9%

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

   3. distribute-neg-frac 93.9%

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

   4. distribute-rgt-neg-out 93.9%

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

   5. associate-/l* 93.9%

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

   6. *-commutative 93.9%

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

   7. *-commutative 93.9%

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

   8. associate-/l* 93.9%

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

3. Simplified 93.9%

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

   1. associate-/l* 93.9%

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

   2. *-commutative 93.9%

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

   3. associate-*l/ 93.9%

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

   4. associate-/l* 98.6%

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

   5. associate-/r/ 98.6%

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

   6. frac-times 98.5%

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

   7. *-commutative 98.5%

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

   8. div-inv 98.4%

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

   9. associate-*l* 98.7%

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

   10. *-commutative 98.7%

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

5. Applied egg-rr 98.7%

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

6. Taylor expanded in sinTheta_i around 0 98.3%

   \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{1}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
7. Step-by-step derivation

   1. associate-/r* 98.5%

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

   2. rec-exp 98.5%

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

   3. distribute-neg-frac 98.5%

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

   4. metadata-eval 98.5%

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

8. Simplified 98.4%

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

9. Final simplification 98.4%

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

Alternative 10: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O}{v \cdot v} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_O (* v v))
  (/ cosTheta_i (- (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 (cosTheta_O / (v * v)) * (cosTheta_i / (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 = (costheta_o / (v * v)) * (costheta_i / (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(cosTheta_O / Float32(v * v)) * Float32(cosTheta_i / 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 = (cosTheta_O / (v * v)) * (cosTheta_i / (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. Step-by-step derivation

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_O around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/l* 98.6%

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

   4. associate-*r/ 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in sinTheta_O around 0 98.4%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
8. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

   2. unpow2 98.4%

      \[\leadsto \frac{cosTheta_O}{\color{blue}{v \cdot v}} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

   3. rec-exp 98.4%

      \[\leadsto \frac{cosTheta_O}{v \cdot v} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}} \]

   4. distribute-neg-frac 98.4%

      \[\leadsto \frac{cosTheta_O}{v \cdot v} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \]

   5. metadata-eval 98.4%

      \[\leadsto \frac{cosTheta_O}{v \cdot v} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

9. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{cosTheta_O}{v \cdot v} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]

10. Final simplification 98.4%

    \[\leadsto \frac{cosTheta_O}{v \cdot v} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]

Alternative 11: 64.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 + \frac{0.3333333333333333}{v \cdot v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ (* cosTheta_i cosTheta_O) (* v (exp (/ (* sinTheta_i sinTheta_O) v))))
  (+ 2.0 (/ 0.3333333333333333 (* v v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / (v * expf(((sinTheta_i * sinTheta_O) / v)))) / (2.0f + (0.3333333333333333f / (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 * costheta_o) / (v * exp(((sintheta_i * sintheta_o) / v)))) / (2.0e0 + (0.3333333333333333e0 / (v * v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / Float32(v * exp(Float32(Float32(sinTheta_i * sinTheta_O) / v)))) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / (v * exp(((sinTheta_i * sinTheta_O) / v)))) / (single(2.0) + (single(0.3333333333333333) / (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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_O around 0 98.6%

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

5. Taylor expanded in v around inf 63.0%

   \[\leadsto \frac{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
6. Step-by-step derivation

   1. associate-*r/ 63.0%

      \[\leadsto \frac{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

   2. metadata-eval 63.0%

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

   3. unpow2 63.0%

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

7. Simplified 63.0%

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

8. Final simplification 63.0%

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

Alternative 12: 58.9% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{cosTheta_O} \cdot \frac{2}{cosTheta_i}} \end{array} \]

FPCore

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

C

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

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_o) * (2.0e0 / costheta_i))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / ((v / cosTheta_O) * (single(2.0) / cosTheta_i));
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. *-commutative 98.6%

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

   2. times-frac 98.5%

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

   3. distribute-neg-frac 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. associate-*l/ 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. *-commutative 57.4%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} \]

   2. associate-/l* 57.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

6. Simplified 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]
7. Step-by-step derivation

   1. associate-*r/ 57.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot cosTheta_i}{\frac{v}{cosTheta_O}}} \]

   2. *-commutative 57.4%

      \[\leadsto \frac{\color{blue}{cosTheta_i \cdot 0.5}}{\frac{v}{cosTheta_O}} \]

   3. metadata-eval 57.4%

      \[\leadsto \frac{cosTheta_i \cdot \color{blue}{\frac{1}{2}}}{\frac{v}{cosTheta_O}} \]

   4. div-inv 57.4%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_i}{2}}}{\frac{v}{cosTheta_O}} \]

   5. clear-num 57.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_O}}{\frac{cosTheta_i}{2}}}} \]

   6. div-inv 57.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{v}{cosTheta_O} \cdot \frac{1}{\frac{cosTheta_i}{2}}}} \]

   7. clear-num 57.7%

      \[\leadsto \frac{1}{\frac{v}{cosTheta_O} \cdot \color{blue}{\frac{2}{cosTheta_i}}} \]

8. Applied egg-rr 57.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta_O} \cdot \frac{2}{cosTheta_i}}} \]

9. Final simplification 57.7%

   \[\leadsto \frac{1}{\frac{v}{cosTheta_O} \cdot \frac{2}{cosTheta_i}} \]

Alternative 13: 58.9% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{2}{cosTheta_O \cdot \frac{cosTheta_i}{v}}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (2.0f / (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 = 1.0e0 / (2.0e0 / (costheta_o * (costheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. times-frac 98.5%

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

   3. distribute-neg-frac 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. associate-*l/ 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. *-commutative 57.4%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} \]

   2. associate-/l* 57.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

6. Simplified 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]
7. Step-by-step derivation

   1. *-commutative 57.4%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot 0.5} \]

   2. div-inv 57.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot 0.5 \]

   3. clear-num 57.4%

      \[\leadsto \left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot 0.5 \]

   4. metadata-eval 57.4%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. div-inv 57.4%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{2}} \]

   6. clear-num 57.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{cosTheta_i \cdot \frac{cosTheta_O}{v}}}} \]

   7. associate-*r/ 57.7%

      \[\leadsto \frac{1}{\frac{2}{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}}} \]

   8. associate-*l/ 57.7%

      \[\leadsto \frac{1}{\frac{2}{\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O}}} \]

   9. *-commutative 57.7%

      \[\leadsto \frac{1}{\frac{2}{\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}}}} \]

8. Applied egg-rr 57.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{2}{cosTheta_O \cdot \frac{cosTheta_i}{v}}}} \]

9. Final simplification 57.7%

   \[\leadsto \frac{1}{\frac{2}{cosTheta_O \cdot \frac{cosTheta_i}{v}}} \]

Alternative 14: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot 0.5 \end{array} \]

FPCore

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

C

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

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / (v / cosTheta_O)) * single(0.5);
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. *-commutative 98.6%

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

   2. times-frac 98.5%

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

   3. distribute-neg-frac 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. associate-*l/ 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. *-commutative 57.4%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} \]

   2. associate-/l* 57.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

6. Simplified 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

7. Final simplification 57.4%

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

Alternative 15: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 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_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_i * costheta_o) / v)
end function

Julia

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. times-frac 98.5%

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

   3. distribute-neg-frac 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. associate-*l/ 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]

5. Final simplification 57.4%

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

Alternative 16: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{0.5}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (0.5f / 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 * costheta_o) * (0.5e0 / v)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) * (single(0.5) / 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-/l* 93.9%

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

   2. *-commutative 93.9%

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

   3. distribute-neg-frac 93.9%

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

   4. distribute-rgt-neg-out 93.9%

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

   5. associate-/l* 93.9%

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

   6. *-commutative 93.9%

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

   7. *-commutative 93.9%

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

   8. associate-/l* 93.9%

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

3. Simplified 93.9%

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

   1. associate-/l* 93.9%

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

   2. *-commutative 93.9%

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

   3. associate-*l/ 93.9%

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

   4. associate-/l* 98.6%

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

   5. associate-/r/ 98.6%

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

   6. frac-times 98.5%

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

   7. *-commutative 98.5%

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

   8. div-inv 98.4%

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

   9. associate-*l* 98.7%

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

   10. *-commutative 98.7%

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

5. Applied egg-rr 98.7%

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

6. Taylor expanded in v around inf 57.4%

   \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{0.5} \]
7. Step-by-step derivation

   1. *-commutative 57.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \]

   2. associate-*r/ 57.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]

   3. clear-num 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}}} \]

   4. un-div-inv 57.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}}} \]

8. Applied egg-rr 57.7%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}}} \]
9. Step-by-step derivation

   1. associate-/r/ 57.4%

      \[\leadsto \color{blue}{\frac{0.5}{v} \cdot \left(cosTheta_i \cdot cosTheta_O\right)} \]

   2. *-commutative 57.4%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(cosTheta_O \cdot cosTheta_i\right)} \]

10. Simplified 57.4%

    \[\leadsto \color{blue}{\frac{0.5}{v} \cdot \left(cosTheta_O \cdot cosTheta_i\right)} \]

11. Final simplification 57.4%

    \[\leadsto \left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{0.5}{v} \]

Alternative 17: 58.9% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / (v / (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 = 0.5e0 / (v / (costheta_i * costheta_o))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v / (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. *-commutative 98.6%

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

   2. times-frac 98.5%

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

   3. distribute-neg-frac 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. associate-*l/ 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. *-commutative 57.4%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} \]

   2. associate-/l* 57.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

6. Simplified 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]
7. Step-by-step derivation

   1. clear-num 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

   2. un-div-inv 57.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

   3. associate-/l/ 57.7%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{v}{cosTheta_i \cdot cosTheta_O}}} \]

   4. *-commutative 57.7%

      \[\leadsto \frac{0.5}{\frac{v}{\color{blue}{cosTheta_O \cdot cosTheta_i}}} \]

8. Applied egg-rr 57.7%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta_O \cdot cosTheta_i}}} \]

9. Final simplification 57.7%

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

Reproduce

herbie shell --seed 2023297 
(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)))