Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 8

Speedup: 1.0×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 60.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+118}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -1.0)
     t_0
     (if (<= z 1.25e-90) x (if (<= z 1.95e+118) (* y z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.25e-90) {
		tmp = x;
	} else if (z <= 1.95e+118) {
		tmp = y * z;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x * -z
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.25d-90) then
        tmp = x
    else if (z <= 1.95d+118) then
        tmp = y * z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.25e-90) {
		tmp = x;
	} else if (z <= 1.95e+118) {
		tmp = y * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 1.25e-90:
		tmp = x
	elif z <= 1.95e+118:
		tmp = y * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.25e-90)
		tmp = x;
	elseif (z <= 1.95e+118)
		tmp = Float64(y * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.25e-90)
		tmp = x;
	elseif (z <= 1.95e+118)
		tmp = y * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 1.25e-90], x, If[LessEqual[z, 1.95e+118], N[(y * z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.95e118 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 59.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 59.5%

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

      2. unsub-neg 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in z around inf 58.7%

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

      1. neg-mul-1 58.7%

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

      2. *-commutative 58.7%

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

   8. Simplified 58.7%

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

   if -1 < z < 1.25000000000000005e-90

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 79.5%

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

   if 1.25000000000000005e-90 < z < 1.95e118

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right) + y \cdot z} \]
   4. Step-by-step derivation

      1. fma-define 100.0%

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

      2. +-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 61.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+118}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.96 \cdot 10^{-17}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.96e-17))) (* (- y x) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.96e-17)) {
		tmp = (y - x) * 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 ((z <= (-1.0d0)) .or. (.not. (z <= 1.96d-17))) then
        tmp = (y - x) * 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 ((z <= -1.0) || !(z <= 1.96e-17)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.96e-17):
		tmp = (y - x) * z
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.96e-17))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.96e-17)))
		tmp = (y - x) * z;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.96e-17]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.96e-17 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 95.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right) + y \cdot z} \]
   4. Step-by-step derivation

      1. fma-define 97.8%

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

      2. +-commutative 97.8%

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

      3. mul-1-neg 97.8%

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

      4. *-commutative 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in z around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

   8. Simplified 99.0%

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

   if -1 < z < 1.96e-17

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.8%

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

      1. *-commutative 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.96 \cdot 10^{-17}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+15} \lor \neg \left(z \leq 5.5 \cdot 10^{-20}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.3e+15) (not (<= z 5.5e-20))) (* (- y x) z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.3e+15) || !(z <= 5.5e-20)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - 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 ((z <= (-4.3d+15)) .or. (.not. (z <= 5.5d-20))) then
        tmp = (y - x) * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.3e+15) || !(z <= 5.5e-20)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.3e+15) or not (z <= 5.5e-20):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.3e+15) || !(z <= 5.5e-20))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.3e+15) || ~((z <= 5.5e-20)))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.3e+15], N[Not[LessEqual[z, 5.5e-20]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.3e15 or 5.4999999999999996e-20 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 95.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right) + y \cdot z} \]
   4. Step-by-step derivation

      1. fma-define 97.7%

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

      2. +-commutative 97.7%

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

      3. mul-1-neg 97.7%

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

      4. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in z around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   8. Simplified 99.7%

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

   if -4.3e15 < z < 5.4999999999999996e-20

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 78.3%

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

      2. unsub-neg 78.3%

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

   5. Simplified 78.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+15} \lor \neg \left(z \leq 5.5 \cdot 10^{-20}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+117} \lor \neg \left(y \leq 1.9 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.9e+117) (not (<= y 1.9e+70))) (* y z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.9e+117) || !(y <= 1.9e+70)) {
		tmp = y * z;
	} else {
		tmp = x * (1.0 - 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 ((y <= (-3.9d+117)) .or. (.not. (y <= 1.9d+70))) then
        tmp = y * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.9e+117) || !(y <= 1.9e+70)) {
		tmp = y * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.9e+117) or not (y <= 1.9e+70):
		tmp = y * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.9e+117) || !(y <= 1.9e+70))
		tmp = Float64(y * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.9e+117) || ~((y <= 1.9e+70)))
		tmp = y * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.9e+117], N[Not[LessEqual[y, 1.9e+70]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8999999999999999e117 or 1.8999999999999999e70 < y

   1. Initial program 99.9%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 93.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right) + y \cdot z} \]
   4. Step-by-step derivation

      1. fma-define 97.4%

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

      2. +-commutative 97.4%

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

      3. mul-1-neg 97.4%

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

      4. *-commutative 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around 0 73.6%

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

   if -3.8999999999999999e117 < y < 1.8999999999999999e70

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 81.0%

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

      2. unsub-neg 81.0%

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

   5. Simplified 81.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+117} \lor \neg \left(y \leq 1.9 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-11} \lor \neg \left(z \leq 9 \cdot 10^{-91}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8e-11) (not (<= z 9e-91))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-11) || !(z <= 9e-91)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	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 ((z <= (-8d-11)) .or. (.not. (z <= 9d-91))) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-11) || !(z <= 9e-91)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8e-11) or not (z <= 9e-91):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8e-11) || !(z <= 9e-91))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8e-11) || ~((z <= 9e-91)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8e-11], N[Not[LessEqual[z, 9e-91]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999952e-11 or 8.99999999999999952e-91 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 95.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right) + y \cdot z} \]
   4. Step-by-step derivation

      1. fma-define 98.0%

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

      2. +-commutative 98.0%

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

      3. mul-1-neg 98.0%

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

      4. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around 0 49.7%

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

   if -7.99999999999999952e-11 < z < 8.99999999999999952e-91

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 80.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-11} \lor \neg \left(z \leq 9 \cdot 10^{-91}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 8: 36.0% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in z around 0 38.2%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024133 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))