Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

Alternatives: 6

Speedup: 1.2×

• 34.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* x (* x x)))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * (x * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * (x * (x * x)))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * (x * (x * x)));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * (x * (x * x)))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(x * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * (x * (x * x)));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

2. Final simplification 99.8%

   \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \]

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.7) (not (<= x 2.7)))
   (* x (* (* x x) -0.12900613773279798))
   (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.7) || !(x <= 2.7)) {
		tmp = x * ((x * x) * -0.12900613773279798);
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.7d0)) .or. (.not. (x <= 2.7d0))) then
        tmp = x * ((x * x) * (-0.12900613773279798d0))
    else
        tmp = 0.954929658551372d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.7) || !(x <= 2.7)) {
		tmp = x * ((x * x) * -0.12900613773279798);
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.7) or not (x <= 2.7):
		tmp = x * ((x * x) * -0.12900613773279798)
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.7) || !(x <= 2.7))
		tmp = Float64(x * Float64(Float64(x * x) * -0.12900613773279798));
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.7) || ~((x <= 2.7)))
		tmp = x * ((x * x) * -0.12900613773279798);
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.7], N[Not[LessEqual[x, 2.7]], $MachinePrecision]], N[(x * N[(N[(x * x), $MachinePrecision] * -0.12900613773279798), $MachinePrecision]), $MachinePrecision], N[(0.954929658551372 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002 or 2.7000000000000002 < x

   1. Initial program 99.9%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

      2. distribute-rgt-out-- 99.9%

         \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

      3. sub-neg 99.9%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\right)} \]

      4. +-commutative 99.9%

         \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right) + 0.954929658551372\right)} \]

      5. *-commutative 99.9%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) + 0.954929658551372\right) \]

      6. distribute-rgt-neg-in 99.9%

         \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798\right)} + 0.954929658551372\right) \]

      7. fma-def 99.9%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

      8. metadata-eval 99.9%

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

   4. Taylor expanded in x around inf 99.3%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 99.3%

         \[\leadsto x \cdot \left(-0.12900613773279798 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   6. Simplified 99.3%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

   if -2.7000000000000002 < x < 2.7000000000000002

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

      2. distribute-rgt-out-- 99.8%

         \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

      3. sub-neg 99.8%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right) + 0.954929658551372\right)} \]

      5. *-commutative 99.8%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) + 0.954929658551372\right) \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798\right)} + 0.954929658551372\right) \]

      7. fma-def 99.8%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

   4. Taylor expanded in x around 0 97.4%

      \[\leadsto x \cdot \color{blue}{0.954929658551372} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)\\ \mathbf{elif}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.7)
   (* x (* (* x x) -0.12900613773279798))
   (if (<= x 2.7)
     (* 0.954929658551372 x)
     (* (* x x) (* x -0.12900613773279798)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = x * ((x * x) * -0.12900613773279798);
	} else if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = (x * x) * (x * -0.12900613773279798);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.7d0)) then
        tmp = x * ((x * x) * (-0.12900613773279798d0))
    else if (x <= 2.7d0) then
        tmp = 0.954929658551372d0 * x
    else
        tmp = (x * x) * (x * (-0.12900613773279798d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = x * ((x * x) * -0.12900613773279798);
	} else if (x <= 2.7) {
		tmp = 0.954929658551372 * x;
	} else {
		tmp = (x * x) * (x * -0.12900613773279798);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.7:
		tmp = x * ((x * x) * -0.12900613773279798)
	elif x <= 2.7:
		tmp = 0.954929658551372 * x
	else:
		tmp = (x * x) * (x * -0.12900613773279798)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.7)
		tmp = Float64(x * Float64(Float64(x * x) * -0.12900613773279798));
	elseif (x <= 2.7)
		tmp = Float64(0.954929658551372 * x);
	else
		tmp = Float64(Float64(x * x) * Float64(x * -0.12900613773279798));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.7)
		tmp = x * ((x * x) * -0.12900613773279798);
	elseif (x <= 2.7)
		tmp = 0.954929658551372 * x;
	else
		tmp = (x * x) * (x * -0.12900613773279798);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.7], N[(x * N[(N[(x * x), $MachinePrecision] * -0.12900613773279798), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7], N[(0.954929658551372 * x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * -0.12900613773279798), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7000000000000002

   1. Initial program 99.9%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

      2. distribute-rgt-out-- 99.9%

         \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

      3. sub-neg 99.9%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\right)} \]

      4. +-commutative 99.9%

         \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right) + 0.954929658551372\right)} \]

      5. *-commutative 99.9%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) + 0.954929658551372\right) \]

      6. distribute-rgt-neg-in 99.9%

         \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798\right)} + 0.954929658551372\right) \]

      7. fma-def 99.9%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

      8. metadata-eval 99.9%

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

   4. Taylor expanded in x around inf 99.8%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto x \cdot \left(-0.12900613773279798 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   6. Simplified 99.8%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

   if -2.7000000000000002 < x < 2.7000000000000002

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

      2. distribute-rgt-out-- 99.8%

         \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

      3. sub-neg 99.8%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right) + 0.954929658551372\right)} \]

      5. *-commutative 99.8%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) + 0.954929658551372\right) \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798\right)} + 0.954929658551372\right) \]

      7. fma-def 99.8%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

   4. Taylor expanded in x around 0 97.4%

      \[\leadsto x \cdot \color{blue}{0.954929658551372} \]

   if 2.7000000000000002 < x

   1. Initial program 99.9%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

      2. distribute-rgt-out-- 99.9%

         \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

      3. sub-neg 99.9%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\right)} \]

      4. +-commutative 99.9%

         \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right) + 0.954929658551372\right)} \]

      5. *-commutative 99.9%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) + 0.954929658551372\right) \]

      6. distribute-rgt-neg-in 99.9%

         \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798\right)} + 0.954929658551372\right) \]

      7. fma-def 99.9%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

      8. metadata-eval 99.9%

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 98.7%

         \[\leadsto x \cdot \left(-0.12900613773279798 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      2. *-commutative 98.7%

         \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)} \]

      3. associate-*r* 98.7%

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)} \]

   6. Simplified 98.7%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 98.8%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)} \]

      2. associate-*r* 98.8%

         \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot -0.12900613773279798} \]

      3. unpow3 98.7%

         \[\leadsto \color{blue}{{x}^{3}} \cdot -0.12900613773279798 \]

      4. *-commutative 98.7%

         \[\leadsto \color{blue}{-0.12900613773279798 \cdot {x}^{3}} \]

      5. add-cube-cbrt 98.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{-0.12900613773279798 \cdot {x}^{3}} \cdot \sqrt[3]{-0.12900613773279798 \cdot {x}^{3}}\right) \cdot \sqrt[3]{-0.12900613773279798 \cdot {x}^{3}}} \]

      6. pow3 98.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.12900613773279798 \cdot {x}^{3}}\right)}^{3}} \]

      7. *-commutative 98.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot -0.12900613773279798}}\right)}^{3} \]

      8. cbrt-prod 98.2%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{-0.12900613773279798}\right)}}^{3} \]

      9. rem-cbrt-cube 98.0%

         \[\leadsto {\left(\color{blue}{x} \cdot \sqrt[3]{-0.12900613773279798}\right)}^{3} \]

   8. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{{\left(x \cdot \sqrt[3]{-0.12900613773279798}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow-prod-down 98.0%

         \[\leadsto \color{blue}{{x}^{3} \cdot {\left(\sqrt[3]{-0.12900613773279798}\right)}^{3}} \]

      2. cube-mult 98.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {\left(\sqrt[3]{-0.12900613773279798}\right)}^{3} \]

      3. rem-cube-cbrt 98.8%

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{-0.12900613773279798} \]

      4. associate-*r* 98.7%

         \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)} \]

      5. *-commutative 98.7%

         \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

      6. add-log-exp 75.3%

         \[\leadsto \color{blue}{\log \left(e^{x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)}\right)} \]

      7. associate-*r* 75.3%

         \[\leadsto \log \left(e^{\color{blue}{\left(x \cdot -0.12900613773279798\right) \cdot \left(x \cdot x\right)}}\right) \]

      8. exp-prod 75.3%

         \[\leadsto \log \color{blue}{\left({\left(e^{x \cdot -0.12900613773279798}\right)}^{\left(x \cdot x\right)}\right)} \]

      9. log-pow 75.6%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \log \left(e^{x \cdot -0.12900613773279798}\right)} \]

      10. add-log-exp 98.8%

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot -0.12900613773279798\right)} \]

   10. Applied egg-rr 98.8%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)\\ \mathbf{elif}\;x \leq 2.7:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.954929658551372 - x \cdot \left(x \cdot 0.12900613773279798\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (- 0.954929658551372 (* x (* x 0.12900613773279798)))))

C

double code(double x) {
	return x * (0.954929658551372 - (x * (x * 0.12900613773279798)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.954929658551372d0 - (x * (x * 0.12900613773279798d0)))
end function

Java

public static double code(double x) {
	return x * (0.954929658551372 - (x * (x * 0.12900613773279798)));
}

Python

def code(x):
	return x * (0.954929658551372 - (x * (x * 0.12900613773279798)))

Julia

function code(x)
	return Float64(x * Float64(0.954929658551372 - Float64(x * Float64(x * 0.12900613773279798))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.954929658551372 - (x * (x * 0.12900613773279798)));
end

Wolfram

code[x_] := N[(x * N[(0.954929658551372 - N[(x * N[(x * 0.12900613773279798), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. associate-*r* 99.8%

      \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

   2. distribute-rgt-out-- 99.8%

      \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

   3. associate-*r* 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\left(0.12900613773279798 \cdot x\right) \cdot x}\right) \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - \left(0.12900613773279798 \cdot x\right) \cdot x\right)} \]

4. Final simplification 99.8%

   \[\leadsto x \cdot \left(0.954929658551372 - x \cdot \left(x \cdot 0.12900613773279798\right)\right) \]

Alternative 5: 51.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot -0.954929658551372\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.7) (* x -0.954929658551372) (* 0.954929658551372 x)))

C

double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = x * -0.954929658551372;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.7d0)) then
        tmp = x * (-0.954929658551372d0)
    else
        tmp = 0.954929658551372d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = x * -0.954929658551372;
	} else {
		tmp = 0.954929658551372 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.7:
		tmp = x * -0.954929658551372
	else:
		tmp = 0.954929658551372 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.7)
		tmp = Float64(x * -0.954929658551372);
	else
		tmp = Float64(0.954929658551372 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.7)
		tmp = x * -0.954929658551372;
	else
		tmp = 0.954929658551372 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.7], N[(x * -0.954929658551372), $MachinePrecision], N[(0.954929658551372 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002

   1. Initial program 99.9%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

      2. distribute-rgt-out-- 99.9%

         \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

      3. sub-neg 99.9%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\right)} \]

      4. +-commutative 99.9%

         \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right) + 0.954929658551372\right)} \]

      5. *-commutative 99.9%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) + 0.954929658551372\right) \]

      6. distribute-rgt-neg-in 99.9%

         \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798\right)} + 0.954929658551372\right) \]

      7. fma-def 99.9%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

      8. metadata-eval 99.9%

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

   4. Taylor expanded in x around 0 0.4%

      \[\leadsto x \cdot \color{blue}{0.954929658551372} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \color{blue}{\sqrt{x \cdot 0.954929658551372} \cdot \sqrt{x \cdot 0.954929658551372}} \]

      2. sqrt-unprod 56.6%

         \[\leadsto \color{blue}{\sqrt{\left(x \cdot 0.954929658551372\right) \cdot \left(x \cdot 0.954929658551372\right)}} \]

      3. swap-sqr 56.6%

         \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(0.954929658551372 \cdot 0.954929658551372\right)}} \]

      4. metadata-eval 56.6%

         \[\leadsto \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.9118906527810399}} \]

   6. Applied egg-rr 56.6%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot x\right) \cdot 0.9118906527810399}} \]

   7. Taylor expanded in x around -inf 6.0%

      \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 6.0%

         \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]

   9. Simplified 6.0%

      \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]

   if -2.7000000000000002 < x

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

      2. distribute-rgt-out-- 99.8%

         \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

      3. sub-neg 99.8%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\right)} \]

      4. +-commutative 99.8%

         \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right) + 0.954929658551372\right)} \]

      5. *-commutative 99.8%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) + 0.954929658551372\right) \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798\right)} + 0.954929658551372\right) \]

      7. fma-def 99.8%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

   4. Taylor expanded in x around 0 62.7%

      \[\leadsto x \cdot \color{blue}{0.954929658551372} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot -0.954929658551372\\ \mathbf{else}:\\ \;\;\;\;0.954929658551372 \cdot x\\ \end{array} \]

Alternative 6: 5.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.954929658551372 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -0.954929658551372))

C

double code(double x) {
	return x * -0.954929658551372;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-0.954929658551372d0)
end function

Java

public static double code(double x) {
	return x * -0.954929658551372;
}

Python

def code(x):
	return x * -0.954929658551372

Julia

function code(x)
	return Float64(x * -0.954929658551372)
end

MATLAB

function tmp = code(x)
	tmp = x * -0.954929658551372;
end

Wolfram

code[x_] := N[(x * -0.954929658551372), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. associate-*r* 99.8%

      \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

   2. distribute-rgt-out-- 99.8%

      \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

   3. sub-neg 99.8%

      \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\right)} \]

   4. +-commutative 99.8%

      \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right) + 0.954929658551372\right)} \]

   5. *-commutative 99.8%

      \[\leadsto x \cdot \left(\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) + 0.954929658551372\right) \]

   6. distribute-rgt-neg-in 99.8%

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(-0.12900613773279798\right)} + 0.954929658551372\right) \]

   7. fma-def 99.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

   8. metadata-eval 99.8%

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, -0.12900613773279798, 0.954929658551372\right)} \]

4. Taylor expanded in x around 0 44.7%

   \[\leadsto x \cdot \color{blue}{0.954929658551372} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 22.3%

      \[\leadsto \color{blue}{\sqrt{x \cdot 0.954929658551372} \cdot \sqrt{x \cdot 0.954929658551372}} \]

   2. sqrt-unprod 29.4%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 0.954929658551372\right) \cdot \left(x \cdot 0.954929658551372\right)}} \]

   3. swap-sqr 29.4%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(0.954929658551372 \cdot 0.954929658551372\right)}} \]

   4. metadata-eval 29.4%

      \[\leadsto \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.9118906527810399}} \]

6. Applied egg-rr 29.4%

   \[\leadsto \color{blue}{\sqrt{\left(x \cdot x\right) \cdot 0.9118906527810399}} \]

7. Taylor expanded in x around -inf 5.2%

   \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
8. Step-by-step derivation

   1. *-commutative 5.2%

      \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]

9. Simplified 5.2%

   \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]

10. Final simplification 5.2%

    \[\leadsto x \cdot -0.954929658551372 \]

Reproduce

herbie shell --seed 2023252 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))