Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

Alternatives: 12

Speedup: 1.0×

Specification

\[0 \leq e \land e \leq 1\]

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin v \cdot \frac{e}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* (sin v) (/ e (+ 1.0 (* e (cos v))))))

C

double code(double e, double v) {
	return sin(v) * (e / (1.0 + (e * cos(v))));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = sin(v) * (e / (1.0d0 + (e * cos(v))))
end function

Java

public static double code(double e, double v) {
	return Math.sin(v) * (e / (1.0 + (e * Math.cos(v))));
}

Python

def code(e, v):
	return math.sin(v) * (e / (1.0 + (e * math.cos(v))))

Julia

function code(e, v)
	return Float64(sin(v) * Float64(e / Float64(1.0 + Float64(e * cos(v)))))
end

MATLAB

function tmp = code(e, v)
	tmp = sin(v) * (e / (1.0 + (e * cos(v))));
end

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * N[(e / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 28.7%

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

   2. *-un-lft-identity 28.7%

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

   3. log-prod 28.7%

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

   4. metadata-eval 28.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{e \cdot \sin v}{1 + e \cdot \cos v}}\right) \]

   5. associate-/l* 28.7%

      \[\leadsto 0 + \log \left(e^{\color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}}}\right) \]

   6. add-log-exp 99.8%

      \[\leadsto 0 + \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]

   7. +-commutative 99.8%

      \[\leadsto 0 + e \cdot \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \]

   8. fma-define 99.8%

      \[\leadsto 0 + e \cdot \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{0 + e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]
5. Step-by-step derivation

   1. +-lft-identity 99.8%

      \[\leadsto \color{blue}{e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

   2. *-commutative 99.8%

      \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e} \]

   3. associate-*l/ 99.8%

      \[\leadsto \color{blue}{\frac{\sin v \cdot e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

   4. associate-*r/ 99.8%

      \[\leadsto \color{blue}{\sin v \cdot \frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

6. Simplified 99.8%

   \[\leadsto \color{blue}{\sin v \cdot \frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

7. Taylor expanded in v around inf 99.8%

   \[\leadsto \sin v \cdot \color{blue}{\frac{e}{1 + e \cdot \cos v}} \]
8. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin v}{\cos v + \frac{1}{e}} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (sin v) (+ (cos v) (/ 1.0 e))))

C

double code(double e, double v) {
	return sin(v) / (cos(v) + (1.0 / e));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = sin(v) / (cos(v) + (1.0d0 / e))
end function

Java

public static double code(double e, double v) {
	return Math.sin(v) / (Math.cos(v) + (1.0 / e));
}

Python

def code(e, v):
	return math.sin(v) / (math.cos(v) + (1.0 / e))

Julia

function code(e, v)
	return Float64(sin(v) / Float64(cos(v) + Float64(1.0 / e)))
end

MATLAB

function tmp = code(e, v)
	tmp = sin(v) / (cos(v) + (1.0 / e));
end

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] / N[(N[Cos[v], $MachinePrecision] + N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 28.7%

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

   2. *-un-lft-identity 28.7%

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

   3. log-prod 28.7%

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

   4. metadata-eval 28.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{e \cdot \sin v}{1 + e \cdot \cos v}}\right) \]

   5. associate-/l* 28.7%

      \[\leadsto 0 + \log \left(e^{\color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}}}\right) \]

   6. add-log-exp 99.8%

      \[\leadsto 0 + \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]

   7. +-commutative 99.8%

      \[\leadsto 0 + e \cdot \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \]

   8. fma-define 99.8%

      \[\leadsto 0 + e \cdot \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{0 + e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]
5. Step-by-step derivation

   1. +-lft-identity 99.8%

      \[\leadsto \color{blue}{e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

   2. *-commutative 99.8%

      \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e} \]

   3. associate-*l/ 99.8%

      \[\leadsto \color{blue}{\frac{\sin v \cdot e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

   4. associate-*r/ 99.8%

      \[\leadsto \color{blue}{\sin v \cdot \frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

6. Simplified 99.8%

   \[\leadsto \color{blue}{\sin v \cdot \frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]
7. Step-by-step derivation

   1. clear-num 99.7%

      \[\leadsto \sin v \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{e}}} \]

   2. un-div-inv 99.7%

      \[\leadsto \color{blue}{\frac{\sin v}{\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{e}}} \]

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sin v}{\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{e}}} \]

9. Taylor expanded in e around inf 99.7%

   \[\leadsto \frac{\sin v}{\color{blue}{\cos v + \frac{1}{e}}} \]
10. Add Preprocessing

Alternative 4: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 10^{-71}:\\ \;\;\;\;\frac{e \cdot v}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 1e-71) (/ (* e v) (+ e 1.0)) (* e (sin v))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 1e-71) {
		tmp = (e * v) / (e + 1.0);
	} else {
		tmp = e * sin(v);
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 1d-71) then
        tmp = (e * v) / (e + 1.0d0)
    else
        tmp = e * sin(v)
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 1e-71) {
		tmp = (e * v) / (e + 1.0);
	} else {
		tmp = e * Math.sin(v);
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 1e-71:
		tmp = (e * v) / (e + 1.0)
	else:
		tmp = e * math.sin(v)
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 1e-71)
		tmp = Float64(Float64(e * v) / Float64(e + 1.0));
	else
		tmp = Float64(e * sin(v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 1e-71)
		tmp = (e * v) / (e + 1.0);
	else
		tmp = e * sin(v);
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 1e-71], N[(N[(e * v), $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 9.9999999999999992e-72

   1. Initial program 99.8%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0 67.4%

      \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

   if 9.9999999999999992e-72 < v

   1. Initial program 99.7%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   2. Add Preprocessing

   3. Taylor expanded in e around 0 98.9%

      \[\leadsto \color{blue}{e \cdot \sin v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 10^{-71}:\\ \;\;\;\;\frac{e \cdot v}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ e 1.0)))

C

double code(double e, double v) {
	return (e * sin(v)) / (e + 1.0);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (e + 1.0d0)
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (e + 1.0);
}

Python

def code(e, v):
	return (e * math.sin(v)) / (e + 1.0)

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(e + 1.0))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (e + 1.0);
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 98.7%

   \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{e}} \]

4. Final simplification 98.7%

   \[\leadsto \frac{e \cdot \sin v}{e + 1} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sin v \cdot \frac{e}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* (sin v) (/ e (+ e 1.0))))

C

double code(double e, double v) {
	return sin(v) * (e / (e + 1.0));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = sin(v) * (e / (e + 1.0d0))
end function

Java

public static double code(double e, double v) {
	return Math.sin(v) * (e / (e + 1.0));
}

Python

def code(e, v):
	return math.sin(v) * (e / (e + 1.0))

Julia

function code(e, v)
	return Float64(sin(v) * Float64(e / Float64(e + 1.0)))
end

MATLAB

function tmp = code(e, v)
	tmp = sin(v) * (e / (e + 1.0));
end

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 28.7%

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

   2. *-un-lft-identity 28.7%

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

   3. log-prod 28.7%

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

   4. metadata-eval 28.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{e \cdot \sin v}{1 + e \cdot \cos v}}\right) \]

   5. associate-/l* 28.7%

      \[\leadsto 0 + \log \left(e^{\color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}}}\right) \]

   6. add-log-exp 99.8%

      \[\leadsto 0 + \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]

   7. +-commutative 99.8%

      \[\leadsto 0 + e \cdot \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \]

   8. fma-define 99.8%

      \[\leadsto 0 + e \cdot \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{0 + e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]
5. Step-by-step derivation

   1. +-lft-identity 99.8%

      \[\leadsto \color{blue}{e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

   2. *-commutative 99.8%

      \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e} \]

   3. associate-*l/ 99.8%

      \[\leadsto \color{blue}{\frac{\sin v \cdot e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

   4. associate-*r/ 99.8%

      \[\leadsto \color{blue}{\sin v \cdot \frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

6. Simplified 99.8%

   \[\leadsto \color{blue}{\sin v \cdot \frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)}} \]

7. Taylor expanded in v around 0 98.7%

   \[\leadsto \sin v \cdot \color{blue}{\frac{e}{1 + e}} \]
8. Step-by-step derivation

   1. +-commutative 98.7%

      \[\leadsto \sin v \cdot \frac{e}{\color{blue}{e + 1}} \]

9. Simplified 98.7%

   \[\leadsto \sin v \cdot \color{blue}{\frac{e}{e + 1}} \]
10. Add Preprocessing

Alternative 7: 51.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot v}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e v) (+ e 1.0)))

C

double code(double e, double v) {
	return (e * v) / (e + 1.0);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * v) / (e + 1.0d0)
end function

Java

public static double code(double e, double v) {
	return (e * v) / (e + 1.0);
}

Python

def code(e, v):
	return (e * v) / (e + 1.0)

Julia

function code(e, v)
	return Float64(Float64(e * v) / Float64(e + 1.0))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * v) / (e + 1.0);
end

Wolfram

code[e_, v_] := N[(N[(e * v), $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 53.5%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]

4. Final simplification 53.5%

   \[\leadsto \frac{e \cdot v}{e + 1} \]
5. Add Preprocessing

Alternative 8: 51.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ v \cdot \frac{e}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* v (/ e (+ e 1.0))))

C

double code(double e, double v) {
	return v * (e / (e + 1.0));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = v * (e / (e + 1.0d0))
end function

Java

public static double code(double e, double v) {
	return v * (e / (e + 1.0));
}

Python

def code(e, v):
	return v * (e / (e + 1.0))

Julia

function code(e, v)
	return Float64(v * Float64(e / Float64(e + 1.0)))
end

MATLAB

function tmp = code(e, v)
	tmp = v * (e / (e + 1.0));
end

Wolfram

code[e_, v_] := N[(v * N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 53.5%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* 53.5%

      \[\leadsto \color{blue}{e \cdot \frac{v}{1 + e}} \]

   2. +-commutative 53.5%

      \[\leadsto e \cdot \frac{v}{\color{blue}{e + 1}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]
6. Step-by-step derivation

   1. clear-num 53.4%

      \[\leadsto e \cdot \color{blue}{\frac{1}{\frac{e + 1}{v}}} \]

7. Applied egg-rr 53.4%

   \[\leadsto e \cdot \color{blue}{\frac{1}{\frac{e + 1}{v}}} \]
8. Step-by-step derivation

   1. un-div-inv 53.4%

      \[\leadsto \color{blue}{\frac{e}{\frac{e + 1}{v}}} \]

   2. clear-num 51.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{e + 1}{v}}{e}}} \]

9. Applied egg-rr 51.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{e + 1}{v}}{e}}} \]
10. Step-by-step derivation

    1. clear-num 53.4%

       \[\leadsto \color{blue}{\frac{e}{\frac{e + 1}{v}}} \]

    2. +-commutative 53.4%

       \[\leadsto \frac{e}{\frac{\color{blue}{1 + e}}{v}} \]

    3. associate-/r/ 53.5%

       \[\leadsto \color{blue}{\frac{e}{1 + e} \cdot v} \]

    4. +-commutative 53.5%

       \[\leadsto \frac{e}{\color{blue}{e + 1}} \cdot v \]

11. Applied egg-rr 53.5%

    \[\leadsto \color{blue}{\frac{e}{e + 1} \cdot v} \]

12. Final simplification 53.5%

    \[\leadsto v \cdot \frac{e}{e + 1} \]
13. Add Preprocessing

Alternative 9: 51.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ e \cdot \frac{v}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (/ v (+ e 1.0))))

C

double code(double e, double v) {
	return e * (v / (e + 1.0));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (v / (e + 1.0d0))
end function

Java

public static double code(double e, double v) {
	return e * (v / (e + 1.0));
}

Python

def code(e, v):
	return e * (v / (e + 1.0))

Julia

function code(e, v)
	return Float64(e * Float64(v / Float64(e + 1.0)))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (v / (e + 1.0));
end

Wolfram

code[e_, v_] := N[(e * N[(v / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 53.5%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* 53.5%

      \[\leadsto \color{blue}{e \cdot \frac{v}{1 + e}} \]

   2. +-commutative 53.5%

      \[\leadsto e \cdot \frac{v}{\color{blue}{e + 1}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]
6. Add Preprocessing

Alternative 10: 50.7% accurate, 29.9× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 (* e (* v (- 1.0 e))))

C

double code(double e, double v) {
	return e * (v * (1.0 - e));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (v * (1.0d0 - e))
end function

Java

public static double code(double e, double v) {
	return e * (v * (1.0 - e));
}

Python

def code(e, v):
	return e * (v * (1.0 - e))

Julia

function code(e, v)
	return Float64(e * Float64(v * Float64(1.0 - e)))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (v * (1.0 - e));
end

Wolfram

code[e_, v_] := N[(e * N[(v * N[(1.0 - e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 53.5%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* 53.5%

      \[\leadsto \color{blue}{e \cdot \frac{v}{1 + e}} \]

   2. +-commutative 53.5%

      \[\leadsto e \cdot \frac{v}{\color{blue}{e + 1}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]

6. Taylor expanded in e around 0 52.5%

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

   1. associate-*r* 52.5%

      \[\leadsto e \cdot \left(v + \color{blue}{\left(-1 \cdot e\right) \cdot v}\right) \]

   2. neg-mul-1 52.5%

      \[\leadsto e \cdot \left(v + \color{blue}{\left(-e\right)} \cdot v\right) \]

   3. *-lft-identity 52.5%

      \[\leadsto e \cdot \left(\color{blue}{1 \cdot v} + \left(-e\right) \cdot v\right) \]

   4. distribute-rgt-in 52.5%

      \[\leadsto e \cdot \color{blue}{\left(v \cdot \left(1 + \left(-e\right)\right)\right)} \]

   5. sub-neg 52.5%

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

8. Simplified 52.5%

   \[\leadsto e \cdot \color{blue}{\left(v \cdot \left(1 - e\right)\right)} \]
9. Add Preprocessing

Alternative 11: 50.1% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ e \cdot v \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e v))

C

double code(double e, double v) {
	return e * v;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * v
end function

Java

public static double code(double e, double v) {
	return e * v;
}

Python

def code(e, v):
	return e * v

Julia

function code(e, v)
	return Float64(e * v)
end

MATLAB

function tmp = code(e, v)
	tmp = e * v;
end

Wolfram

code[e_, v_] := N[(e * v), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 53.5%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* 53.5%

      \[\leadsto \color{blue}{e \cdot \frac{v}{1 + e}} \]

   2. +-commutative 53.5%

      \[\leadsto e \cdot \frac{v}{\color{blue}{e + 1}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]

6. Taylor expanded in e around 0 51.8%

   \[\leadsto e \cdot \color{blue}{v} \]
7. Add Preprocessing

Alternative 12: 4.5% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 v)

C

double code(double e, double v) {
	return v;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = v
end function

Java

public static double code(double e, double v) {
	return v;
}

Python

def code(e, v):
	return v

Julia

function code(e, v)
	return v
end

MATLAB

function tmp = code(e, v)
	tmp = v;
end

Wolfram

code[e_, v_] := v

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0 53.5%

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* 53.5%

      \[\leadsto \color{blue}{e \cdot \frac{v}{1 + e}} \]

   2. +-commutative 53.5%

      \[\leadsto e \cdot \frac{v}{\color{blue}{e + 1}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]

6. Taylor expanded in e around inf 2.4%

   \[\leadsto \color{blue}{v + -1 \cdot \frac{v}{e}} \]
7. Step-by-step derivation

   1. mul-1-neg 2.4%

      \[\leadsto v + \color{blue}{\left(-\frac{v}{e}\right)} \]

   2. unsub-neg 2.4%

      \[\leadsto \color{blue}{v - \frac{v}{e}} \]

8. Simplified 2.4%

   \[\leadsto \color{blue}{v - \frac{v}{e}} \]

9. Taylor expanded in e around inf 4.6%

   \[\leadsto \color{blue}{v} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))