Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3

Percentage Accurate: 97.8% → 100.0%

Time: 4.7s

Alternatives: 8

Speedup: 1.3×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot y + \left(x - 1\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- x 1.0) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + ((x - 1.0d0) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((x - 1.0) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(x - 1.0) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((x - 1.0) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y + \left(x - 1\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- x 1.0) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + ((x - 1.0d0) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((x - 1.0) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(x - 1.0) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((x - 1.0) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y + z, -z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x (+ y z) (- z)))

C

double code(double x, double y, double z) {
	return fma(x, (y + z), -z);
}

Julia

function code(x, y, z)
	return fma(x, Float64(y + z), Float64(-z))
end

Wolfram

code[x_, y_, z_] := N[(x * N[(y + z), $MachinePrecision] + (-z)), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Step-by-step derivation

   1. *-commutative 98.8%

      \[\leadsto x \cdot y + \color{blue}{z \cdot \left(x - 1\right)} \]

   2. sub-neg 98.8%

      \[\leadsto x \cdot y + z \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]

   3. distribute-rgt-in 98.8%

      \[\leadsto x \cdot y + \color{blue}{\left(x \cdot z + \left(-1\right) \cdot z\right)} \]

   4. associate-+r+ 98.8%

      \[\leadsto \color{blue}{\left(x \cdot y + x \cdot z\right) + \left(-1\right) \cdot z} \]

   5. distribute-lft-out 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + z\right)} + \left(-1\right) \cdot z \]

   6. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, \left(-1\right) \cdot z\right)} \]

   7. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(x, y + z, \color{blue}{-1} \cdot z\right) \]

   8. mul-1-neg 100.0%

      \[\leadsto \mathsf{fma}\left(x, y + z, \color{blue}{-z}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, -z\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, y + z, -z\right) \]

Alternative 2: 61.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+149}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+99}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -27:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 820000000000 \lor \neg \left(x \leq 4.5 \cdot 10^{+168}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.32e+149)
   (* x y)
   (if (<= x -2.2e+99)
     (* x z)
     (if (<= x -27.0)
       (* x y)
       (if (<= x 2.5e-62)
         (- z)
         (if (or (<= x 820000000000.0) (not (<= x 4.5e+168)))
           (* x y)
           (* x z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.32e+149) {
		tmp = x * y;
	} else if (x <= -2.2e+99) {
		tmp = x * z;
	} else if (x <= -27.0) {
		tmp = x * y;
	} else if (x <= 2.5e-62) {
		tmp = -z;
	} else if ((x <= 820000000000.0) || !(x <= 4.5e+168)) {
		tmp = x * y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.32d+149)) then
        tmp = x * y
    else if (x <= (-2.2d+99)) then
        tmp = x * z
    else if (x <= (-27.0d0)) then
        tmp = x * y
    else if (x <= 2.5d-62) then
        tmp = -z
    else if ((x <= 820000000000.0d0) .or. (.not. (x <= 4.5d+168))) then
        tmp = x * y
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.32e+149) {
		tmp = x * y;
	} else if (x <= -2.2e+99) {
		tmp = x * z;
	} else if (x <= -27.0) {
		tmp = x * y;
	} else if (x <= 2.5e-62) {
		tmp = -z;
	} else if ((x <= 820000000000.0) || !(x <= 4.5e+168)) {
		tmp = x * y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.32e+149:
		tmp = x * y
	elif x <= -2.2e+99:
		tmp = x * z
	elif x <= -27.0:
		tmp = x * y
	elif x <= 2.5e-62:
		tmp = -z
	elif (x <= 820000000000.0) or not (x <= 4.5e+168):
		tmp = x * y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.32e+149)
		tmp = Float64(x * y);
	elseif (x <= -2.2e+99)
		tmp = Float64(x * z);
	elseif (x <= -27.0)
		tmp = Float64(x * y);
	elseif (x <= 2.5e-62)
		tmp = Float64(-z);
	elseif ((x <= 820000000000.0) || !(x <= 4.5e+168))
		tmp = Float64(x * y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.32e+149)
		tmp = x * y;
	elseif (x <= -2.2e+99)
		tmp = x * z;
	elseif (x <= -27.0)
		tmp = x * y;
	elseif (x <= 2.5e-62)
		tmp = -z;
	elseif ((x <= 820000000000.0) || ~((x <= 4.5e+168)))
		tmp = x * y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.32e+149], N[(x * y), $MachinePrecision], If[LessEqual[x, -2.2e+99], N[(x * z), $MachinePrecision], If[LessEqual[x, -27.0], N[(x * y), $MachinePrecision], If[LessEqual[x, 2.5e-62], (-z), If[Or[LessEqual[x, 820000000000.0], N[Not[LessEqual[x, 4.5e+168]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32000000000000004e149 or -2.19999999999999978e99 < x < -27 or 2.5000000000000001e-62 < x < 8.2e11 or 4.50000000000000012e168 < x

   1. Initial program 96.5%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in y around inf 65.2%

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

   if -1.32000000000000004e149 < x < -2.19999999999999978e99 or 8.2e11 < x < 4.50000000000000012e168

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in y around 0 68.6%

      \[\leadsto \color{blue}{\left(x - 1\right) \cdot z} \]

   3. Taylor expanded in x around inf 67.9%

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

   if -27 < x < 2.5000000000000001e-62

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around 0 75.3%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 75.3%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 75.3%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+149}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+99}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -27:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 820000000000 \lor \neg \left(x \leq 4.5 \cdot 10^{+168}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-7} \lor \neg \left(x \leq 7.5 \cdot 10^{-61}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8e-7) (not (<= x 7.5e-61))) (* x (+ y z)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8e-7) || !(x <= 7.5e-61)) {
		tmp = x * (y + z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8d-7)) .or. (.not. (x <= 7.5d-61))) then
        tmp = x * (y + z)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8e-7) || !(x <= 7.5e-61)) {
		tmp = x * (y + z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8e-7) or not (x <= 7.5e-61):
		tmp = x * (y + z)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8e-7) || !(x <= 7.5e-61))
		tmp = Float64(x * Float64(y + z));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8e-7) || ~((x <= 7.5e-61)))
		tmp = x * (y + z);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8e-7], N[Not[LessEqual[x, 7.5e-61]], $MachinePrecision]], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999996e-7 or 7.50000000000000047e-61 < x

   1. Initial program 97.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around inf 95.6%

      \[\leadsto \color{blue}{\left(y + z\right) \cdot x} \]
   3. Step-by-step derivation

      1. +-commutative 95.6%

         \[\leadsto \color{blue}{\left(z + y\right)} \cdot x \]

   4. Simplified 95.6%

      \[\leadsto \color{blue}{\left(z + y\right) \cdot x} \]

   if -7.9999999999999996e-7 < x < 7.50000000000000047e-61

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around 0 75.8%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 75.8%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 75.8%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-7} \lor \neg \left(x \leq 7.5 \cdot 10^{-61}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -27 \lor \neg \left(x \leq 8.5 \cdot 10^{-62}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -27.0) (not (<= x 8.5e-62))) (* x (+ y z)) (* z (+ x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -27.0) || !(x <= 8.5e-62)) {
		tmp = x * (y + z);
	} else {
		tmp = z * (x + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-27.0d0)) .or. (.not. (x <= 8.5d-62))) then
        tmp = x * (y + z)
    else
        tmp = z * (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -27.0) || !(x <= 8.5e-62)) {
		tmp = x * (y + z);
	} else {
		tmp = z * (x + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -27.0) or not (x <= 8.5e-62):
		tmp = x * (y + z)
	else:
		tmp = z * (x + -1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -27.0) || !(x <= 8.5e-62))
		tmp = Float64(x * Float64(y + z));
	else
		tmp = Float64(z * Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -27.0) || ~((x <= 8.5e-62)))
		tmp = x * (y + z);
	else
		tmp = z * (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -27.0], N[Not[LessEqual[x, 8.5e-62]], $MachinePrecision]], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], N[(z * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -27 or 8.4999999999999995e-62 < x

   1. Initial program 97.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around inf 96.1%

      \[\leadsto \color{blue}{\left(y + z\right) \cdot x} \]
   3. Step-by-step derivation

      1. +-commutative 96.1%

         \[\leadsto \color{blue}{\left(z + y\right)} \cdot x \]

   4. Simplified 96.1%

      \[\leadsto \color{blue}{\left(z + y\right) \cdot x} \]

   if -27 < x < 8.4999999999999995e-62

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in y around 0 76.3%

      \[\leadsto \color{blue}{\left(x - 1\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -27 \lor \neg \left(x \leq 8.5 \cdot 10^{-62}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.000116\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.000116))) (* x (+ y z)) (- (* x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.000116)) {
		tmp = x * (y + z);
	} else {
		tmp = (x * y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.000116d0))) then
        tmp = x * (y + z)
    else
        tmp = (x * y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.000116)) {
		tmp = x * (y + z);
	} else {
		tmp = (x * y) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.000116):
		tmp = x * (y + z)
	else:
		tmp = (x * y) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.000116))
		tmp = Float64(x * Float64(y + z));
	else
		tmp = Float64(Float64(x * y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.000116)))
		tmp = x * (y + z);
	else
		tmp = (x * y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.000116]], $MachinePrecision]], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.16e-4 < x

   1. Initial program 97.5%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around inf 98.0%

      \[\leadsto \color{blue}{\left(y + z\right) \cdot x} \]
   3. Step-by-step derivation

      1. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(z + y\right)} \cdot x \]

   4. Simplified 98.0%

      \[\leadsto \color{blue}{\left(z + y\right) \cdot x} \]

   if -1 < x < 1.16e-4

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x \cdot y + \color{blue}{z \cdot \left(x - 1\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot y + z \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]

      3. distribute-rgt-in 100.0%

         \[\leadsto x \cdot y + \color{blue}{\left(x \cdot z + \left(-1\right) \cdot z\right)} \]

      4. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + x \cdot z\right) + \left(-1\right) \cdot z} \]

      5. metadata-eval 100.0%

         \[\leadsto \left(x \cdot y + x \cdot z\right) + \color{blue}{-1} \cdot z \]

      6. mul-1-neg 100.0%

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

      7. unsub-neg 100.0%

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

      8. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + z\right) - z} \]
   4. Step-by-step derivation

      1. flip-+ 56.5%

         \[\leadsto x \cdot \color{blue}{\frac{y \cdot y - z \cdot z}{y - z}} - z \]

      2. associate-*r/ 54.6%

         \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot y - z \cdot z\right)}{y - z}} - z \]

      3. difference-of-squares 54.7%

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(y + z\right) \cdot \left(y - z\right)\right)}}{y - z} - z \]

      4. sub-neg 54.7%

         \[\leadsto \frac{x \cdot \left(\left(y + z\right) \cdot \color{blue}{\left(y + \left(-z\right)\right)}\right)}{y - z} - z \]

      5. add-sqr-sqrt 25.5%

         \[\leadsto \frac{x \cdot \left(\left(y + z\right) \cdot \left(y + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right)\right)}{y - z} - z \]

      6. sqrt-unprod 54.5%

         \[\leadsto \frac{x \cdot \left(\left(y + z\right) \cdot \left(y + \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right)\right)}{y - z} - z \]

      7. sqr-neg 54.5%

         \[\leadsto \frac{x \cdot \left(\left(y + z\right) \cdot \left(y + \sqrt{\color{blue}{z \cdot z}}\right)\right)}{y - z} - z \]

      8. sqrt-unprod 29.1%

         \[\leadsto \frac{x \cdot \left(\left(y + z\right) \cdot \left(y + \color{blue}{\sqrt{z} \cdot \sqrt{z}}\right)\right)}{y - z} - z \]

      9. add-sqr-sqrt 54.1%

         \[\leadsto \frac{x \cdot \left(\left(y + z\right) \cdot \left(y + \color{blue}{z}\right)\right)}{y - z} - z \]

      10. pow2 54.1%

          \[\leadsto \frac{x \cdot \color{blue}{{\left(y + z\right)}^{2}}}{y - z} - z \]

      11. sub-neg 54.1%

          \[\leadsto \frac{x \cdot {\left(y + z\right)}^{2}}{\color{blue}{y + \left(-z\right)}} - z \]

      12. add-sqr-sqrt 25.0%

          \[\leadsto \frac{x \cdot {\left(y + z\right)}^{2}}{y + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}}} - z \]

      13. sqrt-unprod 53.3%

          \[\leadsto \frac{x \cdot {\left(y + z\right)}^{2}}{y + \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}} - z \]

      14. sqr-neg 53.3%

          \[\leadsto \frac{x \cdot {\left(y + z\right)}^{2}}{y + \sqrt{\color{blue}{z \cdot z}}} - z \]

      15. sqrt-unprod 29.2%

          \[\leadsto \frac{x \cdot {\left(y + z\right)}^{2}}{y + \color{blue}{\sqrt{z} \cdot \sqrt{z}}} - z \]

      16. add-sqr-sqrt 54.7%

          \[\leadsto \frac{x \cdot {\left(y + z\right)}^{2}}{y + \color{blue}{z}} - z \]

   5. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\frac{x \cdot {\left(y + z\right)}^{2}}{y + z}} - z \]
   6. Step-by-step derivation

      1. associate-/l* 56.5%

         \[\leadsto \color{blue}{\frac{x}{\frac{y + z}{{\left(y + z\right)}^{2}}}} - z \]

      2. unpow2 56.5%

         \[\leadsto \frac{x}{\frac{y + z}{\color{blue}{\left(y + z\right) \cdot \left(y + z\right)}}} - z \]

      3. associate-/r* 99.9%

         \[\leadsto \frac{x}{\color{blue}{\frac{\frac{y + z}{y + z}}{y + z}}} - z \]

      4. *-inverses 99.9%

         \[\leadsto \frac{x}{\frac{\color{blue}{1}}{y + z}} - z \]

      5. +-commutative 99.9%

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{z + y}}} - z \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{z + y}}} - z \]

   8. Taylor expanded in z around 0 99.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.000116\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z\\ \end{array} \]

Alternative 6: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -27:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-63}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -27.0) (* x y) (if (<= x 7e-63) (- z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -27.0) {
		tmp = x * y;
	} else if (x <= 7e-63) {
		tmp = -z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-27.0d0)) then
        tmp = x * y
    else if (x <= 7d-63) then
        tmp = -z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -27.0) {
		tmp = x * y;
	} else if (x <= 7e-63) {
		tmp = -z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -27.0:
		tmp = x * y
	elif x <= 7e-63:
		tmp = -z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -27.0)
		tmp = Float64(x * y);
	elseif (x <= 7e-63)
		tmp = Float64(-z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -27.0)
		tmp = x * y;
	elseif (x <= 7e-63)
		tmp = -z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -27.0], N[(x * y), $MachinePrecision], If[LessEqual[x, 7e-63], (-z), N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -27 or 7.00000000000000006e-63 < x

   1. Initial program 97.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in y around inf 54.2%

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

   if -27 < x < 7.00000000000000006e-63

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around 0 75.3%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 75.3%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 75.3%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -27:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-63}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (+ y z)) z))

C

double code(double x, double y, double z) {
	return (x * (y + z)) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y + z)) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) - z;
}

Python

def code(x, y, z):
	return (x * (y + z)) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Step-by-step derivation

   1. *-commutative 98.8%

      \[\leadsto x \cdot y + \color{blue}{z \cdot \left(x - 1\right)} \]

   2. sub-neg 98.8%

      \[\leadsto x \cdot y + z \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]

   3. distribute-rgt-in 98.8%

      \[\leadsto x \cdot y + \color{blue}{\left(x \cdot z + \left(-1\right) \cdot z\right)} \]

   4. associate-+r+ 98.8%

      \[\leadsto \color{blue}{\left(x \cdot y + x \cdot z\right) + \left(-1\right) \cdot z} \]

   5. metadata-eval 98.8%

      \[\leadsto \left(x \cdot y + x \cdot z\right) + \color{blue}{-1} \cdot z \]

   6. mul-1-neg 98.8%

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

   7. unsub-neg 98.8%

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

   8. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto x \cdot \left(y + z\right) - z \]

Alternative 8: 36.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

double code(double x, double y, double z) {
	return -z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -z
end function

Java

public static double code(double x, double y, double z) {
	return -z;
}

Python

def code(x, y, z):
	return -z

Julia

function code(x, y, z)
	return Float64(-z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -z;
end

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 98.8%

   \[x \cdot y + \left(x - 1\right) \cdot z \]

2. Taylor expanded in x around 0 38.8%

   \[\leadsto \color{blue}{-1 \cdot z} \]
3. Step-by-step derivation

   1. mul-1-neg 38.8%

      \[\leadsto \color{blue}{-z} \]

4. Simplified 38.8%

   \[\leadsto \color{blue}{-z} \]

5. Final simplification 38.8%

   \[\leadsto -z \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))